# Additive energy of regular measures in one and higher dimensions, and the
fractal uncertainty principle

Laura Cladek  UCLA Department of Mathematics, Los Angeles, CA 90095-1555.
<EMAIL_ADDRESS>and  Terence Tao  UCLA Department of Mathematics, Los
Angeles, CA 90095-1555<EMAIL_ADDRESS>
###### Abstract.

We obtain new bounds on the additive energy of (Ahlfors-David type) regular
measures in both one and higher dimensions, which implies expansion results
for sums and products of the associated regular sets, as well as more general
nonlinear functions of these sets. As a corollary of the higher-dimensional
results we obtain some new cases of the fractal uncertainty principle in odd
dimensions.

###### 2010 Mathematics Subject Classification:

11B30, 28A80

##  1\. Introduction

This paper is primarily concerned with the estimation of additive energies of
regular measures (in the sense of Ahlfors and David, localized to specific
ranges of scales) on Euclidean spaces $\mathbb{R}^{d}$. We recall the key
definitions. If $x\in\mathbb{R}^{d}$ and $r>0$, we let
$B(x,r)=B^{d}(x,r)\coloneqq\\{y\in\mathbb{R}^{d}:|x-y|\leq r\\}$ denote the
closed ball of radius $r$ centered at $x$, where we use $||$ for the Euclidean
norm on $\mathbb{R}^{d}$.

######  Definition 1.1 (Regular sets).

[2] Let $0\leq\delta\leq d$ with the ambient dimension $d$ an integer, let
$0<\alpha_{0}<\alpha_{1}$ be scales, and let $C\geq 1$. A closed non-empty
subset $X$ of $\mathbb{R}^{d}$ is said to be _$\delta$ -regular_ on scales
$[\alpha_{0},\alpha_{1}]$ with constant $C$ if there exists a Radon
measure111All measures in this paper will be unsigned. $\mu_{X}$ (which we
call a _$\delta$ -regular measure_ or simply _regular measure_ associated to
$X$) obeying the following properties:

  * (i)

$\mu_{X}$ is supported on $X$ (thus $\mu_{X}(\mathbb{R}^{d}\backslash X)=0$);

  * (ii)

For every ball $B(x,r)$ of radius $r\in[\alpha_{0},\alpha_{1}]$, we have the
upper bound $\mu_{X}(B(x,r))\leq Cr^{\delta}$;

  * (iii)

If in addition in (i) we have $x\in X$, then we have the matching lower bound
$\mu_{X}(B(x,r))\geq C^{-1}r^{\delta}$.

Examples of regular sets include Cantor sets, smooth compact submanifolds of
$\mathbb{R}^{d}$, and $r$-neighbourhoods of such sets for $0<r\leq\alpha_{0}$;
see for instance [4] for such examples and further discussion. Now we review
the notion of additive energy at a given scale, as given for instance in [6].

######  Definition 1.2 (Additive energy).

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$, and let $r>0$. The _additive
energy_ ${\operatorname{E}}(\mu,r)$ of $\mu$ at scale $r$ is defined to be the
quantity

${\operatorname{E}}(\mu,r)\coloneqq\mu^{4}\left(\\{(x_{1},x_{2},x_{3},x_{4})\in(\mathbb{R}^{d})^{4}:|x_{1}+x_{2}-x_{3}-x_{4}|\leq
r\\}\right)$ (1.1)

where $\mu^{4}$ is the product measure on $(\mathbb{R}^{d})^{4}$ formed from
four copies of the measure $\mu$ on $\mathbb{R}^{d}$.

By the Fubini–Tonelli theorem one can write

${\operatorname{E}}(\mu,r)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\mu(B(x_{1}+x_{2}-x_{3},r))\
d\mu(x_{1})d\mu(x_{2})d\mu(x_{3})$

so we have the trivial bound

${\operatorname{E}}(\mu,r)\leq\mu(\mathbb{R}^{d})^{3}\sup_{x\in\mathbb{R}^{d}}\mu(B(x,r)).$
(1.2)

In particular, if $\mu=\mu_{X}$ is a regular measure associated to a
$\delta$-regular set on scales $[\alpha,1]$ with constant $C$ that is
supported on the unit ball $B(0,1)$, then we have the “trivial bound”

${\operatorname{E}}(\mu,r)\leq C^{4}r^{\delta}$ (1.3)

for any $\alpha\leq r\leq 1$.

In the case when $\delta$ is an integer, this trivial bound can be sharp up to
multiplicative constants. Indeed, if $\mu=\mu_{X}$ is $\delta$-dimensional
Lebesgue measure on the disk $X\coloneqq
B^{\delta}(0,1)\times\\{0\\}^{d-\delta}$, then for any $0<\alpha\leq r\leq 1$,
one can verify that $\mu$ is regular on scales $[\alpha,1]$ with some
constant222See Section 1.3 for our asymptotic notation conventions.
$C=O_{d,\delta}(1)$, but that
${\operatorname{E}}(\mu,r)\sim_{d,\delta}r^{\delta}$.

However, when the dimension $\delta$ is not an integer one expects an
improvement to the trivial bound (1.3) in the asymptotic regime when $r$ goes
to zero. In one dimension $d=1$ this was achieved by Dyatlov and Zahl [6]:

######  Theorem 1.3 (Improved additive energy bound for regular measures in
$\mathbb{R}$).

[6, Proposition 6.23] Let $0<\delta<1$, $C>1$ and $0<\alpha<1$. Let
$X\subset[0,1]$ be a $\delta$-regular set on scales $[\alpha,1]$, and let
$\mu_{X}$ be an associated regular measure. Then we have

${\operatorname{E}}(\mu,r)\lesssim_{C,\delta}r^{\delta+\beta}$

for any $\alpha\leq r<1$, where $\beta>0$ is of the form

$\beta=\delta\exp\left(-K(1-\delta)^{-14}(1+\log^{14}C)\right)$ (1.4)

for some absolute constant $K$.

Among other things, this theorem was used to establish the first non-trivial
case of the _fractal uncertainty principle_ , which we will return to later in
this introduction. The proof of Theorem 1.3 relied on an induction on scale
argument and some major inverse theorems in additive combinatorics, such as
the Bogulybov–Ruzsa theorem of Sanders [16]. Ultimately, the key point is that
the $\delta$-regular set $X$ is “porous” and cannot exhibit approximate
translation invariance along a medium-length arithmetic progression; on the
other hand, additive combinatorics and induction on scales can be used to
produce such an approximate translation invariance if $\mu$ has exceptionally
high additive energy.

The bound (1.4) on $\beta$ behaves quasipolynomially in $C$. In [6, §6.8.3]
the question was raised as to whether the exponent $\beta$ could be improved
to be polynomial in $C$. Our first main result answers this question in the
affirmative.

######  Theorem 1.4 (Further improvement to additive energy bounds for regular
measures in $\mathbb{R}$).

With the hypotheses of Theorem 1.3, one can take $\beta$ to be of the form

$\beta=c\min(\delta,1-\delta)C^{-25}$

for some absolute constant $c>0$.

The exponent $25$ can be improved here, but we do not attempt to optimize it
in this paper. It may be possible to improve the bound even further; the best
known counterexample (see [6, §6.8.3]) only shows that $\beta$ must decay at
least as fast as $\frac{1}{\log C}$ as $C\to\infty$ (holding $\delta$ fixed).

Our argument is elementary and relies heavily on the order structure of the
real line $\mathbb{R}$, as well as the ability to work with a mesh of dyadic
(or more precisely, $K$-adic for a large $K$) intervals. Roughly speaking, the
idea is to identify a lot of “left333One could also work just as easily with a
notion of “right edge” if desired by switching all the signs. edges” of (a
suitable discretization of) the $\delta$-regular set $X$: ($K$-adic) intervals
$I=[x,x+K^{-2n})$ which intersect the set $X$, but such that there is a large
interval $[x-K^{-2n+1},x)$ immediately to the left of $I$ that is completely
disjoint from $X$. Informally, left edges identify locations and scales where
the set $X$ visibly fails to behave like an additive subgroup of the real
numbers. As $\delta$ is bounded away from $0$ and $1$, we will be able to
exhibit many left edges at many scales, to the point that almost all of the
elements of $X$ will be contained in at least one left edge (or slight
technical generalization of this concept which we call a left near-edge). On
the other hand, if one considers a sum $x_{1}+x_{2}$ arising from a pair
$x_{1},x_{2}\in X$ that are contained in left-edges $[y_{1},y_{1}+K^{-2n})$,
$[y_{2},y_{2}+K^{-2n})$ respectively, then there are unusually few other pairs
$x_{3},x_{4}\in X$ with $x_{3},x_{4}$ close (within $K^{-2n+1}/10$ say) of
$x_{1},x_{2}$ respectively, such that $x_{1}+x_{2}\approx x_{3}+x_{4}$. This
is because in the vicinity of the left-edge $[y_{1},y_{1}+K^{-2n})$, most of
the candidates for $x_{3}$ lie far to the right of $x_{1}$, and similarly most
of the candidates of $x_{4}$ lie far to the right of $x_{2}$, so it is
difficult to keep $x_{3}+x_{4}$ close to $x_{1}+x_{2}$; see Figure 1. Because
of this, each pair of left-edges can be used to create a slight diminution of
the additive energy; by combining the effect of all the available pairs of
left-edges at all scales, we can obtain a preliminary improvement to the
trivial bound (1.3) on the additive energy at some fixed small scale $r_{0}$
(made precise in Proposition 2.1 below), which can then be iterated by a
standard “induction on scales” argument to produce Theorem 1.4. Our bounds are
superior to that in Theorem 1.3 because they do not rely on the
Bogulybov–Ruzsa theorem of Sanders [16], for which the best known bounds are
only quasipolynomial in nature.

Figure 1. The $\delta$-regular set $X$ can be covered by intervals of length
$K^{-2n}$ for some given $n$, which are arranged in a fractal-type pattern.
Some of these intervals are depicted here, on two different portions of the
real line $\mathbb{R}$. The intervals $[y_{1},y_{1}+K^{-2n})$ and
$[y_{2},y_{2}+K^{-2n})$ are left edges: they have nearby intervals covering
$X$ to the right, but not to the left. Note that if $x_{1}\in
X\cap[y_{1}+y_{1}+K^{-2n})$ and $x_{4}\in X\cap[y_{4},y_{4}+K^{-2n})$ then
there are a relatively small number of additional pairs $(x_{3},x_{4})$ of
real numbers $x_{3},x_{4}\in X$ that are somewhat close to $x_{1},x_{2}$
respectively such that $x_{1}+x_{2}\approx x_{3}+x_{4}$. This leads to a small
but non-zero diminuition of the additive energy at the scales between to
$K^{-2n}$ and $K^{-2n+1}$, which one can then hope to iterate.

Our second result addresses the higher dimensional case. Here the order-
theoretic arguments do not seem to be effective (except possibly in the very
high-dimensional regime $d-1<\delta<d$), and we revert to using more tools
from additive combinatorics, and in particular the Bogulybov–Ruzsa theorem. As
such, our bounds are of the same general quasipolynomial shape as in Theorem
1.3, though it seems likely that further progress on the “polynomial
Freiman–Ruzsa conjecture” (see e.g., [17]) may eventually lead to polynomial
bounds along the lines of Theorem 1.4.

######  Theorem 1.5 (Improved additive energy bound for higher dimensional
regular measures).

Let $d\geq 1$ be an integer, let $0<\delta<d$ be a non-integer, and let $C>1$
and $0<\alpha<1$. Let $X\subset B^{d}(0,1)$ be a $\delta$-regular set on
scales $[\alpha,1]$, and let $\mu=\mu_{X}$ be an associated regular measure.
Then we have

${\operatorname{E}}(\mu,r)\lesssim_{C,\delta,d}r^{\delta+\beta}$

for any $\alpha\leq r<1$, where $\beta=\beta_{d,\delta,C}>0$ takes the
quasipolynomial form

$\beta=\exp\left(-O_{\delta,d}(1+\log^{O_{\delta,d}(1)}(C))\right).$

We prove this theorem in Section 5. As with Theorem 1.4, it suffices to gain a
small amount over the trivial bound at some fixed scale $r_{0}$ as one can
then use induction on scales to then obtain the power saving for arbitrary
scales $r$. Our proofs shares some features in common with that in [6], in
that one assumes that the energy is unusually large and uses this to locate a
non-trivial arithmetic progression along which $\mu$ obeys an approximate
symmetry. If the dimension $\delta$ was less than $1$ then one could use the
regularity to obtain a contradiction (basically because a $\delta$-dimensional
measure cannot support an entire line segment if $\delta<1$). The main novelty
in our argument is the treatment of the higher dimensional case $\delta>1$.
Here the strategy is to “quotient out” by the arithmetic progression and
obtain (at least on some range of scales) what is (morally at least) a
$\delta-1$-regular measure supported on a $d-1$-dimensional hyperplane of
$\mathbb{R}^{d}$. This can then be treated by a suitable induction hypothesis
(note how the crucial hypothesis that $\delta$ is not an integer is propagated
by this process). In order to make the notion of “quotienting out” by an
arithmetic progression (which is merely an approximate subgroup of
$\mathbb{R}^{d}$, as opposed to a genuine subgroup) rigorous, we will rely
heavily on the machinery of additive combinatorics, most notably the theory of
the Gowers uniformity norm $U^{2}$ and of approximate groups, and how one can
split up the problem of estimating a global $U^{2}$ Gowers norm into the “fine
scale” problem of estimating various local Gowers norms restricted to a
“coset” $4H+y$ of some approximate group $H$, and the “coarse scale” problem
of controlling the $U^{2}$ norm of the output of these local Gowers norms
(viewed as a function of $y$); see Lemma 4.8(iii) for a precise statement.

######  Remark 1.6.

It seems likely that the techniques in this paper can be combined with those
in [14] to obtain new $L^{q}$ improving bounds for convolutions of regular
measures, but we will not pursue this direction here.

One can use these additive energy bounds to obtain expansion estimates for
both linear and nonlinear maps. Here is a sample such result. For any
measurable $E\subset\mathbb{R}^{d}$ we let $|E|$ denote its Lebesgue measure.

######  Theorem 1.7 (Nonlinear expansion).

Let $d\geq 1$ be an integer, let $0<\delta<d$ be a non-integer, and let $C>1$
and $0<r<1$. Let $F:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a
$C^{2}$ (twice continuously differentiable) map such that for any $x,y\in
B(0,2)$, the derivative maps
$D_{x}F(x,y),D_{y}F(x,y):\mathbb{R}^{d}\to\mathbb{R}^{d}$ are invertible. Let
$X,Y\subset B(0,1)$ be $\delta$-regular sets on scales $[r,1]$, and let
$X_{r}\coloneqq X+B(0,r)$, $Y_{r}\coloneqq Y+B(0,r)$ be the $r$-neighborhoods
of $X,Y$ respectively. Then the set

$F(X_{r},Y_{r})\coloneqq\\{F(x,y):x\in X_{r},y\in Y_{r}\\}$

has measure

$|F(X_{r},Y_{r})|\gtrsim_{C,\delta,d,F}r^{d-\delta-\beta}$

where $\beta$ takes the form

$\beta=\exp\left(-O_{\delta,d,F}(1+\log^{O_{\delta,d}(1)}(C))\right).$

When $d=1$, we can take

$\beta=c_{F}\min(\delta,1-\delta)C^{-25}$

for some $c_{F}>0$ depending on $F$.

In particular, we have

$|X_{r}+X_{r}|,|X_{r}-X_{r}|\gtrsim_{C,\delta,d}r^{d-\delta-\beta}$ (1.5)

and in one dimension we similarly have

$|X_{r}\cdot X_{r}|\gtrsim_{C,\delta}r^{1-\delta-\beta}$ (1.6)

if $X$ lies in (say) $[1/2,1]$ (this can be deduced either by taking
logarithms, or by applying the above theorem to the multiplication map
$x,y\mapsto xy$ after applying some mild changes of variable to avoid the
degeneracies at $x,y=0$). Here we use the usual sumset notations

$\displaystyle A+B$ $\displaystyle\coloneqq\\{a+b:a\in A,b\in B\\}$
$\displaystyle A-B$ $\displaystyle\coloneqq\\{a-b:a\in A,b\in B\\}$
$\displaystyle A+b$ $\displaystyle\coloneqq\\{a+b:a\in A\\}$ $\displaystyle-A$
$\displaystyle\coloneqq\\{-a:a\in A\\}$ $\displaystyle A\cdot B$
$\displaystyle\coloneqq\\{ab:a\in A,b\in B\\}.$

For comparison, it is easy to see from Definition 1.1 that

$|X_{r}|\sim_{C,\delta,d}r^{d-\delta}$

so the bounds in (1.5), (1.6) represent a power gain over the trivial bound
$|A+B|\geq|A|,|B|$ (and $|A\cdot B|\gtrsim|A|,|B|$ when $A,B\subset[1,2]$).

We establish Theorem 1.7 in Section 6. This result can be compared with the
celebrated discretized sum-product theorem of Bourgain [1], which in our
language would give a bound of the form

$\max(|X_{r}+X_{r}|,|X_{r}\cdot X_{r}|)\gtrsim_{C,\delta}r^{\delta+\beta}$
(1.7)

for some $\beta=\beta_{\delta}>0$ and $X\subset[1,2]$, but where the lower
bound in Definition 1.1(iii) is only assumed to hold at scale $r=1$; see the
recent preprint [10] for an explicit value of $\beta$ as a function of
$\delta$. In this more general set up there are standard constructions (in
which $X$ resembles either a long arithmetic progression or a long geometric
progression at various scales) that show that one can no longer expect the
separate bounds (1.5), (1.6), and and can only hope for (1.7). This is
consistent with results such as (1.5), (1.6) because regular sets cannot
support long arithmetic progressions or long geometric progressions. Under the
same general hypotheses on $X$ considered in the above-mentioned result (1.7)
of Bourgain, an expansion bound of the form

$|F(X_{r},X_{r})|\gtrsim_{C,\delta,F}r^{\delta+\beta}$

was also recently established in [13] for polynomial maps $F$ under the
(necessary) Elekes-Rónyai condition that $F(x,y)$ is not of the form
$h(a(x)+b(y))$ or $h(a(x)b(y))$ for some polynomials $h,a,b$.

###  1.1. Application to the fractal uncertainty principle

Let $d\geq 1$ be a dimension, and let $0<h\leq 1$ be a small parameter. We
then define the semiclassical Fourier transform ${\mathcal{F}}_{h}f\colon
L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{d})$ by the formula

${\mathcal{F}}_{h}f(\xi)\coloneqq(2\pi
h)^{-d/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi/h}f(x)\ dx$

for Schwartz functions $f$, extended to $L^{2}(\mathbb{R}^{d})$ by continuity
in the usual fashion. The _fractal uncertainty principle_ concerns operator
norm estimates of the form

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\lesssim h^{\sigma}$

for various sets $X_{h},Y_{h}$ (which are permitted to depend on $h$) and
various exponents $\sigma$. Here and in the sequel we use $1_{E}$ to denote
the indicator function of a set $E$. A model case is when $X,Y$ are
$\delta$-regular subsets of $B(0,1)$ at scales $[h,1]$ for some constant $C$,
and $X_{h}\coloneqq X+B(0,h)$, $Y_{h}\coloneqq Y+B(0,h)$ are the
$h$-neighborhoods of $X,Y$ respectively. A standard application of
Plancherel’s theorem and trivial bound

$\|{\mathcal{F}}_{h}f\|_{L^{\infty}(\mathbb{R}^{d})}\lesssim_{d}h^{-d/2}\|f\|_{L^{1}(\mathbb{R}^{d})}$

then gives the “trivial bound”

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\lesssim_{C,d}h^{\max\left(\frac{d}{2}-\delta,0\right)}$
(1.8)

(see e.g., [4] for the argument in the one-dimensional case $d=1$, which
extends without difficulty to higher dimensions). In one dimension $d=1$ with
$0<\delta<1$, the fractal uncertainty principle [6], [5], [2] asserts that one
can improve this bound to

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\lesssim_{C,\delta}h^{\max\left(\frac{1}{2}-\delta,0\right)+\beta}$
(1.9)

for some $\beta>0$ depending only on $C,\delta$. Indeed, the following
specific values for $\beta$ are known:

  * (i)

[6, Theorems 4, 6] One can take

$\beta=\frac{3}{8}\left(\frac{1}{2}-\delta\right)-\max\left(\frac{1}{2}-\delta,0\right)+\frac{1}{16}\delta\exp(-K(1-\delta)^{14}(1+\log^{14}C))$
(1.10)

for some absolute constant $K>0$ (this only gives a positive value of $\beta$
for $\delta$ sufficiently close to $1/2$).

  * (ii)

[12, Theorem 1.2] For $\delta\geq 1/2$, one can take

$\beta=\exp\left[-\exp(K(C\delta^{-1}(1-\delta)^{-1})^{K(1-\delta)^{-2}})\right]$
(1.11)

for an absolute constant $K>0$.

fractal uncertainty principles can be applied to quantum chaos to obtain lower
bounds on the mass of eigenfunctions and to produce spectral gaps on various
negatively curved manifolds see [4] for a survey of this connection. fractal
uncertainty principles have also been established for a wider class of sets
than regular sets, and particular to _porous sets_ , but we will not discuss
this generalization further here. We remark that while the exponent (1.10) was
obtained via the additive energy method, other fractal uncertainty principle
estimates, such as the one giving (1.11), relied on somewhat different
techniques, such as the Beurling–Malliavin multiplier theorem.

By combining [6, Theorem 4] (or [4, Proposition 5.4]) with Theorem 1.4, we can
immediately improve (1.10) to

$\beta=\frac{3}{8}\left(\frac{1}{2}-\delta\right)-\max\left(\frac{1}{2}-\delta,0\right)+c\min(\delta,1-\delta)C^{-25}$

for an absolute constant $c>0$; this is still only an improvement over the
trivial bound for $\delta$ very close to $1/2$, but the dependence on the
regularity constant $C$ has improved.

In higher dimension $d>1$, a similar combination of [6, Theorem 4] and Theorem
1.5 gives the estimate

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\lesssim_{C,d,\delta}h^{\max(\frac{d}{2}-\delta,0)+\beta}$

with

$\beta=\frac{3}{8}\left(\frac{d}{2}-\delta\right)-\max\left(\frac{d}{2}-\delta,0\right)+\exp\left(-O_{\delta,d}(1+\log^{O_{\delta,d}(1)}(C))\right)$
(1.12)

when $\delta$ is a non-integer. In the case of even dimension this does
not444In particular, our results do not directly make progress on [4,
Conjecture 6.2], which concerns the two-dimensional situation $d=2$. give a
positive value of $\beta$ for any value of $\delta$, which is to be expected
since the no improvement to the trivial exponent $\max(\frac{d}{2}-\delta,0)$
is possible at the critical value $\delta=d/2$ in this case (see [4, Example
6.1] for a counterexample when $d=2$, and a similar construction also works in
other even dimensions). However when $d$ is odd, $d/2$ is a non-integer, and
for $\delta$ sufficiently close to $d/2$ the implied constants in (1.12) can
then be verified to be uniform in $\delta$ (for $d$ fixed), leading to a
fractal uncertainty principle of the form

$\beta=\frac{3}{8}\left(\frac{d}{2}-\delta\right)-\max\left(\frac{d}{2}-\delta,0\right)+\exp\left(-O_{d}(1+\log^{O_{d}(1)}C)\right)$

which is non-negative for $\delta$ sufficiently close to $d/2$. To the authors
knowledge, this is the first higher dimensional fractal uncertainty principle
that holds for _arbitrary_ regular sets. In the case when one of the sets
$X,Y$ was the Cartesian product of $d$ one-dimensional regular (or porous)
sets, a higher dimensional fractal uncertainty principle (with exponents
similar in shape to (1.11)) was recently obtained in [11]. Also, as observed
in [4, §VI.A], a higher dimensional fractal uncertainty principle can
sometimes be deduced from iterating the one-dimensional principle if certain
projections and fibers of $X,Y$ are assumed to be porous.

If one allows the sets $X,Y$ to be regular with different dimensional
parameters $\delta,\delta^{\prime}$ then the additive energy bounds in
Theorems 1.4, 1.5 can give further fractal uncertainty principles. Indeed we
have the following statement:

######  Theorem 1.8 (Consequence of additive energy bounds).

Let $d\geq 1$ be an integer, let $0<\delta<d$ be a non-integer, and let $C>1$
and $0<h<1$. Let $Y\subset B^{d}(0,1)$ be a $\delta$-regular set on scales
$[h,1]$ with constant $C$. Then

$\|{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R}^{d})\to
L^{8}(\mathbb{R}^{d})}\lesssim_{C,d,\delta}h^{-3\delta/8+\beta}$ (1.13)

where

$\beta=\exp\left(-O_{\delta,d}(1+\log^{O_{\delta,d}(1)}(C))\right);$

in the $d=1$ case one can instead take

$\beta=c\min(\delta,1-\delta)C^{-25}$

for some absolute constant $c>0$. In particular, by Hölder’s inequality, if
$X\subset B^{d}(0,1)$ is $\delta^{\prime}$-regular on scales $[h,1]$ with
constant $C^{\prime}$ for some $0\leq\delta^{\prime}\leq d$ and
$C^{\prime}>1$, then

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\lesssim_{C,C^{\prime},d,\delta,\delta^{\prime}}h^{\frac{3}{8}(d-\delta-\delta^{\prime})+\beta}$
(1.14)

We prove this theorem in Section 7. One can compare (1.14) against the trivial
bound, which in this case is

$\|1_{X_{h}}{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\lesssim_{C,C^{\prime},d,\delta,\delta^{\prime}}h^{\max\left(\frac{d-\delta-\delta^{\prime}}{2},0\right)}.$
(1.15)

Thus one has new fractal uncertainty principles for $\delta^{\prime}$-regular
$X$ and $\delta$-regular $Y$ when $\delta$ is non-integer and
$\delta+\delta^{\prime}$ is sufficiently close to $d$; by duality one also has
similar results when it is $\delta^{\prime}$ that is assumed to be non-integer
in place of $\delta$. As before, a modification of [4, Example 6.1] shows that
no improvement of the trivial bound (1.15) can be obtained when
$\delta,\delta^{\prime}$ are both integers.

###  1.2. Acknowledgments

The first author is supported by a National Science Foundation Postdoctoral
Fellowship, NSF grant 1703715. The second author is supported by NSF grant
DMS-1764034 and by a Simons Investigator Award. The first author would also
like to thank Semyon Dyatlov for helpful conversations, and Josh Zahl for a
correction.

###  1.3. Notation

We use the asymptotic notation $X\lesssim Y$, $Y\gtrsim X$, or $X=O(Y)$ to
denote the bound $|X|\leq CY$ for an absolute constant $C>0$, and use $X\sim
Y$ synonymously with $X\lesssim Y\lesssim X$. If we need the constant $C$ to
depend on additional parameters (e.g., the dimension $d$), we indicate this by
subscripts, thus for instance $X\lesssim_{d}Y$ denotes the bound $|X|\leq
C_{d}Y$ for some constant $C_{d}>0$ that depends on $d$.

If $E$ is a finite set, we use $\\#E$ to denote its cardinality.

##  2\. The one-dimensional case: obtaining a slight gain

In this section we begin the proof of Theorem 1.4. Namely, we establish the
following seemingly weaker variant in which one obtains only a slight
improvement over the trivial bound (1.3), but at a fixed scale $r_{0}>0$. In
the next section we will use standard induction on scales argument to iterate
this slight improvement to a power gain in the scale parameter.

######  Proposition 2.1 (Slight gain over the trivial bound in one dimension).

Let $0<\delta<1$, $C>1$ and $0<\varepsilon\leq 1/2$. Let $r_{0}>0$ be the
quantity

$r_{0}\coloneqq\exp\left(-C_{2}\frac{C^{16}\log^{2}(C/\varepsilon)}{\varepsilon^{2}\min(\delta,1-\delta)}\right)$
(2.1)

for a sufficiently large absolute constant $C_{2}$. Let $X\subset\mathbb{R}$
be a $\delta$-regular set on scales $[r_{0},1]$ with constant $C$, and let
$\mu=\mu_{X}$ be an associated regular measure. Then we have

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\leq\varepsilon r_{0}^{\delta}.$

We now turn to the proof of this proposition. Let $0<\delta<1$, $C>1$,
$0<\varepsilon\leq 1/2$, and define $r_{0}$ by (2.1). We let
$X\subset\mathbb{R}^{d}$ be $\delta$-regular on scales $[r_{0},1]$ with
constant $C$ and associated regular measure $\mu_{X}$. Our task is to
establish the bound

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\leq\varepsilon r_{0}^{\delta}.$

We first estimate the left-hand side by an integral involving the
$\mu^{2}$-measure of various “strips” $S_{z}$:

######  Lemma 2.2.

One has

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\lesssim
r_{0}^{-1}\int_{[-3,3]}\mu^{2}(S_{z})^{2}\ dz$ (2.2)

where $\mu^{2}=\mu\times\mu$ is the product of two copies of the measure
$\mu$, and for each $z\in\mathbb{R}$, $S_{z}$ denotes the strip

$S_{z}\coloneqq\\{(x,y)\in[-1,1]^{2}:|x+y-z|\leq r_{0}\\}$

(see Figure 2).

Figure 2. A strip $S_{z}$.

###### Proof.

By the Fubini–Tonelli theorem, the right-hand side is

$r_{0}^{-1}\int_{\mathbb{R}^{4}}\left(\int_{[-3,3]}1_{|x_{1}+x_{4}-z|,|x_{2}+x_{3}-z|\leq
r_{0}}\ dz\right)\ d\mu^{4}(x_{1},x_{2},x_{3},x_{4}).$

If $x_{1},x_{2},x_{3},x_{4}\in[-1,1]$ are such that
$|x_{1}-x_{2}-x_{3}+x_{4}|\leq r_{0}$, then direct calculation shows that

$\int_{[-3,3]}1_{|x_{1}+x_{4}-z|,|x_{2}+x_{3}-z|\leq r_{0}}\ dz\gtrsim r_{0}$

and the claim follows. ∎

To estimate the right-hand side of (2.2) we shall remove a small exceptional
set $E$ from $[-1,1]^{2}$:

######  Corollary 2.3.

For any measurable subset $E$ of $[-1,1]^{2}$, one has

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\lesssim
C^{3}r_{0}^{\delta}\left(\mu^{2}(E)+\sup_{z\in[-3,3]}\mu(\pi(X^{2}\cap
S_{z}\backslash E))\right)$

where $\pi\colon\mathbb{R}^{2}\to\mathbb{R}$ is the projection to the first
coordinate: $\pi(x,y)\coloneqq x$.

###### Proof.

Since $\mu^{2}$ is supported on $X^{2}$, we can split

$\mu^{2}(S_{z})^{2}\leq\mu^{2}(S_{z}\cap E)\mu^{2}(S_{z})+\mu^{2}(X^{2}\cap
S_{z}\backslash E)\mu^{2}(S_{z})$

and thus

$\displaystyle{\operatorname{E}}(\mu|_{[-1,1]},r_{0})$ $\displaystyle\lesssim
r_{0}^{-1}\int_{[-3,3]}\mu^{2}(S_{z}\cap E)\mu^{2}(S_{z})\ dz$
$\displaystyle\quad+r_{0}^{-1}\left(\sup_{z\in[-3,3]}\mu^{2}(X^{2}\cap
S_{z}\backslash E)\right)\int_{[-3,3]}\mu^{2}(S_{z})\ dz.$

By the Fubini–Tonelli theorem one has

$\displaystyle\int_{[-3,3]}\mu^{2}(S_{z})\ dz$
$\displaystyle=\int_{[-1,1]^{2}}\int_{[-3,3]}1_{|x+y-z|\leq r_{0}}\
dzd\mu^{2}(x,y)$ $\displaystyle\lesssim r_{0}\mu([-1,1])^{2}$
$\displaystyle\lesssim C^{2}r_{0}$

and similarly

$\displaystyle\int_{-3}^{3}\mu^{2}(S_{z}\cap E)\mu^{2}(S_{z})\ dz$
$\displaystyle=\int_{E}\int_{[-1,1]^{2}}\int_{-3}^{3}1_{|x_{1}+x_{4}-z|,|x_{2}+x_{3}-z|\leq
r_{0}}\ dzd\mu^{2}(x_{2},x_{3})d\mu^{2}(x_{1},x_{4})$ $\displaystyle\lesssim
r_{0}\int_{E}\int_{[-1,1]}\mu(B(x_{1}-x_{2}+x_{4},2r_{0}))\
d\mu(x_{2})d\mu^{2}(x_{1},x_{4})$ $\displaystyle\lesssim
r_{0}\int_{E}\mu([-1,1])Cr_{0}^{\delta}\ d\mu^{2}(x_{1},x_{4})$
$\displaystyle\lesssim C^{2}r_{0}^{1+\delta}\mu^{2}(E).$

Finally, for any $z$, one has from the Fubini–Tonelli theorem again that

$\displaystyle\mu^{2}(X^{2}\cap S_{z}\backslash E)$
$\displaystyle\leq\int_{\pi(X^{2}\cap S_{z}\backslash
E)}\mu([x-r_{0},x+r_{0}])\ d\mu(x)$ $\displaystyle\lesssim
Cr_{0}^{\delta}\mu(\pi(X^{2}\cap S_{z}\backslash E)).$

Combining all these estimates, we obtain the claim. ∎

To construct this exceptional set $E$ we will take advantage of the porous
nature of the set $X$ as viewed through a sparse $K$-adic mesh. Let $K\geq
1000$ be a perfect square and multiple of $100$ to be chosen later, chosen so
large that

$K^{\delta/2},K^{(1-\delta)/2}\geq C_{0}C^{2}$ (2.3)

for some large absolute constant $C$. Define a _$K$ -adic interval_ to be a
half-open interval of the form $I=[jK^{-n},(j+1)K^{-n})$ for some integers
$j,n$. Such an interval will be said to be _active_ if it intersects $X$, and
_inactive_ otherwise. We have the following basic porosity property:

######  Lemma 2.4 (Porosity).

There does not exist a sequence $I_{1},\dots,I_{\sqrt{K}}$ of $\sqrt{K}$
consecutive active $K$-adic intervals of equal length in $[r_{0},1]$.

###### Proof.

Suppose for contradiction that there was a sequence $I_{1},\dots,I_{\sqrt{K}}$
of consecutive active $K$-adic intervals of some length $K^{-n}\in[r_{0},1]$.
From the regularity of $\mu$, we have

$\mu(I_{j}+[-K^{-n},K^{-n}])\gtrsim C^{-1}K^{-n\delta}$

for all $j=1,\dots,\sqrt{K}$, hence on summing and noting the bounded overlap
of the $I_{j}+[-K^{-n},K^{-n}]$

$\mu\left(\bigcup_{j=1}^{\sqrt{K}}I_{j}+[-K^{-n},K^{-n}]\right)\gtrsim
C^{-1}\sqrt{K}K^{-n\delta}.$

On the other hand, from regularity again we have

$\mu\left(\bigcup_{j=1}^{\sqrt{K}}I_{j}+[-K^{-n},K^{-n}]\right)\lesssim
CK^{(1/2-n)\delta}.$

This contradicts (2.3) if $C_{0}$ is large enough. ∎

Now we make some further definitions concerning $K$-adic intervals (see Figure
3):

  * (i)

If $I=[jK^{-n},(j+1)K^{-n})$ is a $K$-adic interval, we define the _right
shift_ of $I$ to be the $K$-adic interval
$I^{+}\coloneqq[(j+1)K^{-n},(j+2)K^{-n})$, the _left shift_ of $I$ to be the
$K$-adic interval $I^{-}\coloneqq[(j-1)K^{-n},jK^{-n})$, and the _parent_
$I^{*}$ to be the unique $K$-adic interval of $I$ of length $K^{1-n}$ that
contains $I$. We then call $I$ a _child_ of $I^{*}$, and define a _grandchild_
to be a child of a child.

  * (ii)

Given two intervals $I,J$, we say that $I$ _lies to the left_ of $J$, or
equivalently that $J$ _lies to the right_ of $I$, if $x<y$ for all $x\in I$
and $y\in J$.

  * (iii)

A _sibling_ of a $K$-adic interval $I$ is another $k$-adic interval $J$ with
the same parent as $I$: $I^{*}=J^{*}$. A _left sibling_ (resp. right sibling)
of $I$ is a sibling $J$ that lies to the left (resp. right) of $I$.

  * (iv)

A _left edge_ is an active $K$-adic interval $I$ of length
$K^{-2n}\in[r_{0},1]$ for some $n\geq 1$, with the property that all left
siblings of $I$, as well as the left shift $(I^{*})^{-}$ of $I$, are inactive.

  * (v)

A _left near-edge_ is a $K$-adic interval $J$, to which there is associated a
left edge $I$ of the same length as $J$ and equal to or to the left of $J$,
such that all the $K$-adic intervals of the same length as $J$ between $J$ and
$I$ are active. (In particular, if $I$ is a left edge, then $I$ and $I^{+}$
are left near-edges, and from Lemma 2.4 each left edge is associated to at
most $K/100$ left near-edges, which are all adjacent to each other and of the
same length as the left edge, with the rightmost of these left near-edges
being inactive.

Figure 3. Some selected $K$-adic intervals (with $K=4$). The interval $J$ here
is a grandchild of $I^{*}$, and its parent $J^{*}$ is a left sibling of $I$.
If the intervals in red are active and the intervals in black are inactive,
and $I$ has length $K^{-2n}\in[r_{0},1]$ for some $n\geq 1$, then $I$ will be
a left edge and $I,I^{+},I^{++}$ will be left near-edges, but none of the
other intervals of length $K^{-2n}$ depicted here will be left near-edges.

One could also define the notion of a right edge and right near-edge, but we
will not need to do so here.

######  Example 2.5.

Let $K=10$, and let $X\subset[0,1]$ be the set of all real numbers whose
decimal expansions take values in the set $\\{1,2,5,6\\}$. Then an $10$-adic
interval of length $10^{-2n}$ is an interval of the form
$[\frac{a}{10^{2n}},\frac{a+1}{10^{2n}})$, where $a$ is an integer. This
interval is active if $a$ is positive with decimal expansion length $2n$ and
has all digits in $\\{1,2,5,6\\}$; a left-edge arises if furthermore the final
digit is $1$ and the penultimate digit lies in $\\{1,5\\}$. Finally, a left
near-edge of length $10^{-2n}$ arises if $a$ is positive with decimal
expansion of length $2n$ with the final digit in $\\{1,2,3\\}$, penultimate
digit in $\\{1,5\\}$, and all other digits in $\\{1,2,5,6\\}$. Observe that
almost all elements of $X$ will lie in at least one left near-edge, in the
sense that the set of exceptions has strictly smaller dimension than $X$
itself. This phenomenon of abundance of left near-edges is crucial to our
argument.

We make the following basic observations. Firstly, if $I$ is a left edge, then
it is active, and from the regularity property of $\mu$ we have

$\mu(I^{-}\cup I\cup I^{+})\gtrsim C^{-1}|I|^{\delta}.$

On the other hand, as $I$ is a left edge, $I^{-}$ cannot be active (it is
either a left sibling of $I$, or lies in $(I^{*})^{-}$) and thus

$\mu(I\cup I^{+})\gtrsim C^{-1}|I|^{\delta}.$ (2.4)

Thus at least one of the left near-edges $I,I^{+}$ associated to $I$ must
absorb a relatively large amount of the mass of $\mu$ (compared to the upper
bound of $C|I|^{\delta}$ coming from the regularity hypothesis).

Next, we claim that any two left edges $I,J$ of the same length $|I|=|J|$ must
be separated from each other by at least $K|I|$. Indeed, we may assume without
loss of generality that $J$ lies to the left of $I$; as the left siblings of
$I$, as well as the entirety of $(I^{*})^{-}$, are inactive, this forces $J$
to lie to the left of $(I^{*})^{-}$, giving the claim. In particular, we see
that the left near-edges associated to $I$ are disjoint from the left near-
edges associated to $J$.

Now we make the key claim that lets us produce many left edges at many scales.

######  Lemma 2.6 (Many left edges).

Let $n\geq 1$ be such that $K^{-2n}\in[K^{2}r_{0},1]$. Let $I$ be an active
$K$-adic interval of length $K^{-2n}$. Then $I\cup I^{-}$ contains a left edge
$J$ of length $K^{-2}|I|=K^{-2(n+1)}$. Furthermore, if $J$ is associated to a
left near-edge $J^{\prime}$ that is contained in a larger left near-edge
$\tilde{J}$, then $I$ is also contained in a left-near edge of the same length
as $\tilde{J}$.

Figure 4. A typical situation that arises in the proof of Lemma 2.6 (only a
portion of the large intervals $\tilde{J},\tilde{J}^{+}$ are depicted here).
Active intervals are displayed in red, inactive intervals in black. In this
example $J,J^{\prime}$ is contained in $I^{-}$, but it is also possible for
one or both of these intervals to lie in $I$ instead. Similarly, it is also
possible for $I$ to lie in $\tilde{J}$ rather than $\tilde{J}^{+}$. Note that
$J$ is a left edge.

###### Proof.

By Lemma 2.4, at least one of the children $I^{\prime}$ of $I^{-}$ will be
inactive. On the other hand, by the pigeonhole principle at least one of the
grandchildren of $I$ is active. In particular we can find an active interval
$J$ of length $K^{-2}|I|$ that lies to the right of $I^{\prime}$, is a
grandchild of either $I$ or $I^{-}$, and is the leftmost interval with these
properties; see Figure 4. By Lemma 2.4 $J$ lies at a distance of at most
$\sqrt{K}|I^{\prime}|=K^{-1/2}|I^{\prime}|$ of $I^{\prime}$, and thus is a
grandchild of either $I$ or $I^{\prime}$.

By construction, all the intervals of length $K^{-2}|I|$ between $I^{\prime}$
and $J$ are inactive. As $I^{\prime}$ is also inactive, this makes $J$ a left
edge by definition. Now suppose that there is a left near-edge $J^{\prime}$
associated to $J$ is contained in a larger left near-edge $\tilde{J}$, thus
$\tilde{J}$ is at least as large as $I$. Note that $J^{\prime}$ is either
equal to $J$, or lies to the right at a distance of at most
$\sqrt{K}|J|=K^{-3/2}|I|$ thanks to Lemma 2.4; in particular, $J^{\prime}$
also lies in $I^{-}\cup I$. If $J^{\prime}$ lies in $I$ then $\tilde{J}$ will
contain $I$ and we are done, so suppose $J^{\prime}$ lies in $I^{-}$. Then
$\tilde{J}$ contains $I^{-}$, hence contain $J$, hence is active, hence
$\tilde{J}^{+}$ is also a left near-edge. As $\tilde{J}$ contains $I^{-}$, $I$
will be contained in either $\tilde{J}$ or $\tilde{J}^{+}$, and the claim
follows. ∎

Now we can construct the exceptional set $E\subset[-1,1]^{2}$. Let $N$ be the
largest integer such that $K^{-2N}\geq r_{0}$. For any $0\leq n\leq N$, we let
$E_{n}$ be the set of all elements of $[-1,1]^{2}$ that do not lie in a square
$I_{1}\times I_{2}$, where $I_{1},I_{2}$ are two left near-edges of equal
length $K^{-2n^{\prime}}$ for some $1\leq n^{\prime}\leq n$. Clearly
$E_{0}=[-1,1]^{2}$, hence by regularity

$\mu^{2}(E_{0})\lesssim C^{2}.$

Thanks to Lemma 2.6, we can now show a geometric decrease in the measures of
the $E_{n}$:

######  Proposition 2.7 (Geometric decrease).

For every $0\leq n\leq N-1$, one has

$\mu^{2}(E_{n+1})\leq(1-cC^{-4}K^{-4\delta})\mu^{2}(E_{n})$

for some absolute constant $c>0$.

###### Proof.

It suffices to show that

$\mu^{2}(E_{n}\backslash E_{n+1})\gtrsim C^{-4}K^{-4\delta}\mu^{2}(E_{n}).$

By construction, $E_{n}$ is the union of squares $I_{1}\times I_{2}$, where
$I_{1},I_{2}$ are $K$-adic intervals of length $K^{-2n}$ such that there is no
$1\leq n^{\prime}\leq n$ for which $I_{1},I_{2}$ are respectively contained in
two left-near edges $\tilde{I}_{1},\tilde{I}_{2}$ of length
$K^{-2n^{\prime}}$. If $I_{1}$ or $I_{2}$ are inactive then the square
$I_{1}\times I_{2}$ has zero measure in $\mu$, so we can restrict attention to
squares $I_{1}\times I_{2}$ which are _active_ in the sense that $I_{1}$ and
$I_{2}$ are both active. By regularity, the contribution of each active square
can be bounded by

$\mu^{2}(I_{1}\times
I_{2})\lesssim(C|I_{1}|^{\delta})(C|I_{2}|^{\delta})=C^{2}K^{-4n\delta}.$

Thus, if there are $M$ active squares, we have

$\mu^{2}(E_{n})\lesssim C^{2}K^{-4n\delta}M$

and so it will now suffice to establish the bound

$\mu^{2}(E_{n}\backslash E_{n+1})\gtrsim C^{-2}K^{-4n\delta-4\delta}M.$

From Lemma 2.6, for each active square $I_{1}\times I_{2}$ one can find left
edges $J_{1}\subset I_{1}^{-}\cup I_{1}$, $J_{2}\subset I_{2}^{-}\cup I_{2}$
of length $K^{-2n-2}$ obeying the conclusions of the lemma. From (2.4) and the
pigeonhole principle we can find for each $i=1,2$ an interval $J^{\prime}_{i}$
that is either equal to $J_{i}$ or its right shift $J_{i}^{+}$, such that

$\mu(J^{\prime}_{i})\gtrsim C^{-1}(K^{-2n-2})^{\delta}$

and hence

$\mu^{2}(J^{\prime}_{1}\times J^{\prime}_{2})\gtrsim
C^{-2}K^{-4n\delta-4\delta}.$

Note that $J^{\prime}_{1},J^{\prime}_{2}$ are left near-edges associated to
$J_{1},J_{2}$ respectively. In particular, from Lemma 2.6, since $I_{1},I_{2}$
fail to be respectively contained in left-near edges of length
$K^{-2n^{\prime}}$ for some $1\leq n^{\prime}\leq n$, the same is true for
$J^{\prime}_{1},J^{\prime}_{2}$. By construction of $E_{n+1}$, this implies
that

$J^{\prime}_{1}\times J^{\prime}_{2}\subset E_{n}\backslash E_{n+1}.$

Since $J^{\prime}_{1}\times J^{\prime}_{2}$ lies within $O(K^{-2n})$ of
$I_{1}\times I_{2}$, we see that each square $J^{\prime}_{1}\times
J^{\prime}_{2}$ can be generated by at most $O(1)$ of the $M$ active squares
$I_{1}\times I_{2}$. Thus we have

$\mu^{2}(E_{n}\backslash E_{n+1})\gtrsim C^{-2}K^{-4n\delta-4\delta}M$

giving the claim. ∎

Iterating this proposition we see that

$\displaystyle\mu^{2}(E)$ $\displaystyle\lesssim
C^{2}(1-cC^{-4}K^{-4\delta})^{N}$ $\displaystyle\lesssim
C^{2}\exp(-cNC^{-4}K^{-4\delta})$ $\displaystyle\lesssim
C^{2}r_{0}^{\frac{c^{\prime}}{C^{4}K^{4\delta}\log K}}$

for some absolute constants $c,c^{\prime}>0$.

Now let $z\in[-3,3]$. In view of Corollary 2.3, we are interested in bounding
the expression $\mu(\pi(X^{2}\cap S_{z}\backslash E)))$. We can of course
write this as $\mu(Z)$, where $Z$ is the set

$Z\coloneqq\pi(X^{2}\cap S_{z}\backslash E).$

By construction of $E$, we can then bound

$\mu(Z)\leq\sum_{I_{1},I_{2}}\mu\left(\pi\left(X^{2}\cap
S_{z}\cap\left(\bigcup_{I^{\prime}_{1}}I^{\prime}_{1}\times\bigcup_{I^{\prime}_{2}}I_{2}\right)\right)\right)$

where $I_{1},I_{2}$ range over pairs of left-edges of equal length in
$[r_{0},1]$, and $I^{\prime}_{1},I^{\prime}_{2}$ range over the left near-
edges associated with $I_{1},I_{2}$ respectively. Now we make a key
calculation.

######  Lemma 2.8 (Bounding a piece of $\mu(Z)$).

To every pair $I_{1},I_{2}$ of left edges of equal length in $[r_{0},1]$, one
can find a set $Y_{I_{1},I_{2}}\subset[-2,2]$ such that

$\mu\left(\pi\left(X^{2}\cap
S_{z}\cap\left(\bigcup_{I^{\prime}_{1}}I^{\prime}_{1}\times\bigcup_{I^{\prime}_{2}}I_{2}\right)\right)\right)\lesssim
C^{2}K^{-\delta/2}\mu(Y_{I_{1},I_{2}}).$ (2.5)

Furthermore, the sets $Y_{I_{1},I_{2}}$ are disjoint as $(I_{1},I_{2})$ vary.

The key points here are the gain of $K^{-\delta/2}$ on the right-hand side,
and the disjointness of the sets $Y_{I_{1},I_{2}}$.

Figure 5. A typical situation that arises in the proof of Lemma 2.8. To reduce
clutter we take $I_{1}=I^{\prime}_{1}$ and $I_{2}=I^{\prime}_{2}$. The set
$X^{2}$ (which supports the measure $\mu^{2}$) is contained in the squares
formed by products of active intervals, such as $I_{1}\times I_{2}$ (which is
a typical component of the complement of $E$). Because $I_{1},I_{2}$ are left-
edges, if $S_{z}$ intersects $I^{\prime}_{1}\times I^{\prime}_{2}=I_{1}\times
I_{2}$, then the set $\pi(X^{2}\cap S_{z}\cap(I^{\prime}_{1}\times
I^{\prime}_{2}))$, being contained in $I_{1}=I^{\prime}_{1}$, will be much
smaller than the nearby $Y_{I_{1},I_{2}}$ as measured using the regular
measure $\mu$; on the other hand, $Y_{I_{1},I_{2}}$ stays at a medium distance
from the set $Z=\pi(X^{2}\cap S_{z}\backslash E)$ and from $I_{1}$, which will
ensure that the $Y_{I_{1},I_{2}}$ are disjoint as $I_{1},I_{2}$ vary. Note how
this picture resembles the Cartesian product of the two portions of the real
line depicted in Figure 1.

###### Proof.

Write $I_{1}=[x_{1},x_{1}+K^{-2n})$ and $I_{2}=[x_{2},x_{2}+K^{-2n}]$ for some
$K^{-2n}\in[r_{0},1]$. We can assume that the set $X^{2}\cap
S_{z}\cap(\bigcup_{I^{\prime}_{1}}I^{\prime}_{1}\times\bigcup_{I^{\prime}_{2}}I_{2})$
is non-empty, otherwise we can simply set $Y_{I_{1},I_{2}}$ to be the empty
set. From Lemma 2.4, the intervals $I^{\prime}_{i}$ lie within
$2\sqrt{K}K^{-2n}$ of $x_{i}$ for $i=1,2$, hence by definition of $S_{z}$ we
have

$|z-x_{1}-x_{2}|\leq 5\sqrt{K}K^{-2n}.$ (2.6)

also we see that from the definition of $Z$ that $Z$ contains a point within
$2\sqrt{K}K^{-2n}$ of $x_{1}$, thus

$\mathrm{dist}(x_{1},Z)\leq 2\sqrt{K}K^{-2n}.$ (2.7)

We define $Y_{I_{1},I_{2}}$ to be the interval

$Y_{I_{1},I_{2}}\coloneqq\left[x_{1}+100\sqrt{K}K^{-2n},x_{1}+\frac{1}{100}K^{-2n+1}\right];$
(2.8)

see Figure 5. Clearly $Y_{I_{1},I_{2}}\subset[-2,2]$. We now verify (2.5). We
can use regularity to bound

$\displaystyle\mu\left(\pi\left(S_{z}\cap\left(\bigcup_{I^{\prime}_{1}}I^{\prime}_{1}\times\bigcup_{I^{\prime}_{2}}I_{2}\right)\right)\right)$
$\displaystyle\leq\mu\left(\bigcup_{I^{\prime}_{1}}I^{\prime}_{1}\right)$
$\displaystyle\lesssim C(\sqrt{K}K^{-2n})^{\delta}.$

Next, as $I_{1}$ is a left edge, it contains an element $x_{*}$ of $X$, and

$\mu([x_{1}-K^{-2n+1},x_{1}))=0$

and hence

$\mu(Y_{I_{1},I_{2}})\geq\mu\left(B(x_{*},\frac{1}{200}K^{-2n+1})\backslash
B(x_{*},200\sqrt{K}K^{-2n})\right).$

From regularity we have

$\mu\left(B(x_{*},\frac{1}{200}K^{-2n+1})\right)\gtrsim
C^{-1}K^{(-2n+1)\delta}$

and

$\mu\left(B(x_{*},200\sqrt{K}K^{-2n})\right)\lesssim CK^{(-2n+1/2)\delta}$

hence by (2.3) we have (for $C_{0}$ large enough) that

$\mu(Y_{I_{1},I_{2}})\gtrsim C^{-1}K^{(-2n+1)\delta}$

and (2.5) follows.

It remains to establish the disjointness of the $Y_{I_{1},I_{2}}$. To do this
we study the relative position of $Y_{I_{1},I_{2}}$ and $Z$. From (2.7) one
has

$\mathrm{dist}(y,Z)\leq\frac{1}{10}K^{-2n+1}$ (2.9)

for all $y\in Y_{I_{1},I_{2}}$. Now we use the fact that $I_{2}$ is a left-
edge, thus

$X\cap[x_{2}-K^{-2n+1},x_{2})=\emptyset.$

By definition of $S_{z}$, this implies that the set $Z\subset\pi(X^{2}\cap
S_{z})$ avoids the interval $(z-x_{2}+r_{0},z-x_{2}+K^{-2n+1}-r_{0})$. Using
(2.6), we conclude that $Z$ avoids the interval

$\left[x_{1}+10\sqrt{K}K^{-2n},x_{1}+\frac{1}{2}K^{-2n+1}\right]$

(say), and thus we have

$\mathrm{dist}(y,Z)\geq 10K^{-2n+1/2}$ (2.10)

for all $y\in Y_{I_{1},I_{2}}$.

The bounds (2.9), (2.10) imply that two intervals
$Y_{I_{1},I_{2}},Y_{I^{\prime}_{1},I^{\prime}_{2}}$ cannot overlap if the
lengths $|I_{1}|=|I_{2}|=K^{-2n}$,
$|I^{\prime}_{1}|=|I^{\prime}_{2}|=K^{-2n^{\prime}}$ are distinct. Thus it
remains to show that the intervals
$Y_{I_{1},I_{2}},Y_{I^{\prime}_{1},I^{\prime}_{2}}$ cannot overlap in the case
of all equal lengths
$|I_{1}|=|I_{2}|=|I^{\prime}_{1}|=|I^{\prime}_{2}|=K^{-2n}$. Recall that if
the left-edges $I_{1},I^{\prime}_{1}$ are distinct, they are separated from
each other by at least $K^{-2n+1}$. Comparing this with (2.8) we see that one
can only have $Y_{I_{1},I_{2}},Y_{I^{\prime}_{1},I^{\prime}_{2}}$ intersect if
$I_{1}=I^{\prime}_{1}$. But then from (2.6) we see that left-edges
$I_{2},I^{\prime}_{2}$ are separated by at most $10\sqrt{K}K^{-2n}$, and hence
must also be equal from the separation property of left-edges. Thus we see
that the $Y_{I_{1},I_{2}}$ are indeed disjoint as $(I_{1},I_{2})$ vary. ∎

Summing the above lemma over all pairs $(I_{1},I_{2})$ we conclude that

$\mu(Z)\lesssim C^{2}K^{-\delta/2}\mu([-2,2])\lesssim C^{3}K^{-\delta/2}$

and hence by Corollary 2.3 we have

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\lesssim
C^{6}r_{0}^{\delta}\left(r_{0}^{\frac{c^{\prime}}{C^{4}K^{4\delta}\log
K}}+K^{-\delta/2}\right).$

If we now set

$K\coloneqq(C_{1}C^{6}/\varepsilon)^{\max(\frac{2}{\delta},\frac{2}{1-\delta})}$

for a sufficiently large absolute constant $C_{1}$, the the condition (2.3)
will be satisfied, and

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\leq
r_{0}^{\delta}\left(\frac{\varepsilon}{2}+O\left(C^{6}r_{0}^{\frac{c^{\prime\prime}\varepsilon^{2}\min(\delta,1-\delta)}{C^{16}\log(C/\varepsilon)}}\right)\right)$

for some absolute constant $c>0$. Thus if we select

$r_{0}\coloneqq\exp\left(-C_{2}\frac{C^{16}\log^{2}(C/\varepsilon)}{\varepsilon^{2}\min(\delta,1-\delta)}\right)$

for a sufficiently large absolute constant $C_{2}$, we obtain

${\operatorname{E}}(\mu|_{[-1,1]},r_{0})\leq\varepsilon r_{0}^{\delta}$

as required. This concludes the proof of Proposition 2.1.

##  3\. The one-dimensional case: induction on scales

In this section we show how one can iterate Proposition 2.1 to obtain Theorem
1.4.

Let $\delta,C,\alpha,X,\mu_{X}$ be as in Theorem 1.4. Let $\varepsilon>0$ be a
small parameter to be chosen later, and let $r_{0}$ be the quantity in
Proposition 2.1.

It is convenient to adopt the notation of Gowers uniformity norms [9]. For
$f_{1},f_{2},f_{3},f_{4}\in L^{4/3}(\mathbb{R})$, define the Gowers inner
product

$\langle
f_{1},f_{2},f_{3},f_{4}\rangle_{U^{2}(\mathbb{R})}\coloneqq\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}f_{1}(x)\overline{f_{2}}(x+h)\overline{f_{3}}(x+k)f_{4}(x+h+k)\
dxdhdk$

and the Gowers uniformity norm

$\|f\|_{U^{2}(\mathbb{R})}\coloneqq\langle f,f,f,f\rangle^{1/4}$

for any $f\in L^{4/3}(\mathbb{R})$. One can also write the $U^{2}$ norm in
terms of the Fourier transform as

$\|f\|_{U^{2}(\mathbb{R})}=\|\hat{f}\|_{L^{4}(\mathbb{R})}$ (3.1)

so it is clear that the $U^{2}(\mathbb{R})$ norm is indeed a norm. We also
recall the well-known _Gowers–Cauchy–Schwarz inequality_

$|\langle
f_{1},f_{2},f_{3},f_{4}\rangle_{U^{2}(\mathbb{R})}|\leq\|f_{1}\|_{U^{2}(\mathbb{R})}\|f_{2}\|_{U^{2}(\mathbb{R})}\|f_{3}\|_{U^{2}(\mathbb{R})}\|f_{4}\|_{U^{2}(\mathbb{R})}$
(3.2)

We can relate the Gowers norms to additive energy as follows:

######  Lemma 3.1 (Additive energy and Gowers norms).

If $\mu$ is a finite Radon measure on $\mathbb{R}$ and $r>0$, then

${\operatorname{E}}(\mu,r)\sim r^{-3}\|\mu*1_{[-r,r]}\|_{U^{2}}^{4}$ (3.3)

(compare with Lemma 2.2). Also, for any $\lambda>0$ one has

$\|\mu*1_{[-\lambda r,\lambda
r]}\|_{U^{2}}\sim_{\lambda}\|\mu*1_{[-r,r]}\|_{U^{2}}.$ (3.4)

###### Proof.

The claim (3.4) is clear after observing that $1_{[-\lambda r,\lambda_{r}]}$
can be bounded by the sum of $O_{\lambda}(1)$ translates of $1_{[-r,r]}$ (and
vice versa), together with the triangle inequality and translation invariance
of Gowers norms.

By the Fubini–Tonelli theorem we have

$\|\mu*1_{[-r,r]}\|_{U^{2}}^{4}=\int_{\mathbb{R}^{4}}1_{[-r,r]}*1_{[-r,r]}*1_{[-r,r]}*1_{[-r,r]}(x_{1}+x_{2}-x_{3}-x_{4})\
d\mu(x_{1})\dots d\mu(x_{4}).$

Since

$1_{[-r,r]}*1_{[-r,r]}*1_{[-r,r]}*1_{[-r,r]}\gtrsim r^{3}1_{[-r,r]}$

we obtain the bound

$\|\mu*1_{[-r,r]}\|_{U^{2}}^{4}\gtrsim r^{3}{\operatorname{E}}(\mu,r).$

Conversely, since

$1_{[-r/4,r/4]}*1_{[-r/4,r/4]}*1_{[-r/4,r/4]}*1_{[-r/4,r/4]}\lesssim
r^{3}1_{[-r,r]}$

we have

$\|\mu*1_{[-r/4,r/4]}\|_{U^{2}}^{4}\lesssim r^{3}{\operatorname{E}}(\mu,r)$

and the claim (3.3) now follows from (3.4). ∎

The key claim is the following inequality relating the energy at different
scales:

######  Proposition 3.2 (Energy at nearby scales).

If $\alpha\leq r\leq r_{0}$, one has

${\operatorname{E}}(\mu,r)\lesssim C^{4}\varepsilon
r_{0}^{\delta}{\operatorname{E}}(\mu,r/r_{0})$

###### Proof.

By Lemma 3.1 one has

${\operatorname{E}}(\mu,r)\sim
r^{-3}\|\mu*1_{[-r,r]}\|_{U^{2}(\mathbb{R})}^{4}.$

Partition $\mathbb{R}$ into a collection ${\mathcal{I}}$ of half-open
intervals $I$ of length $r/r_{0}$, then we can decompose

$\mu=\sum_{I}\mu|_{I}$

where $\mu_{I}$ denotes the restriction of $\mu$ to $I$, and thus

${\operatorname{E}}(\mu,r)\sim\sum_{I_{1},I_{2},I_{3},I_{4}}r^{-3}\langle\mu|_{I_{1}}*1_{[-r,r]},\dots,\mu|_{I_{4}}*1_{[-r,r]}\rangle_{U^{2}(\mathbb{R})}$

where $I_{1},\dots,I_{4}$ are understood to vary over ${\mathcal{I}}$. The
integral vanishes unless $I_{1}-I_{2}-I_{3}+I_{4}$ intersects $[-4r,4r]$ and
all of the $\mu|_{I_{i}}$ are non-vanishing, which by Definition 1.1(iii)
implies that

$\mu(2I_{i}-I_{i})\gtrsim C^{-1}(r/r_{0})^{\delta},$

where $2I_{i}-I_{i}=I_{i}+[-r/r_{0},r/r_{0}]$ is the triple of $I_{i}$. When
this occurs, we can use the Gowers–Cauchy–Schwarz inequality to estimate

$\langle\mu|_{I_{1}}*1_{[-r,r]},\dots,\mu|_{I_{4}}*1_{[-r,r]}\rangle_{U^{2}(\mathbb{R})}\leq\prod_{i=1}^{4}\|\mu|_{I_{i}}*1_{[-r,r]}\|_{U^{2}(\mathbb{R})}$

and hence by Lemma 3.1 again

$r^{-3}\langle\mu|_{I_{1}}*1_{[-r,r]},\dots,\mu|_{I_{4}}*1_{[-r,r]}\rangle_{U^{2}(\mathbb{R})}\lesssim\prod_{i=1}^{4}{\operatorname{E}}(\mu|_{I_{i}},r)^{1/4}.$

For each $i$, let $T_{i}\colon\mathbb{R}\to\mathbb{R}$ be the affine (order-
preserving) map that sends $I_{i}$ to $[-1,1]$. Direct calculations shows that
the pushforward measure $(r_{0}/r)^{\delta}T_{*}\mu$ is $\delta$-regular at
scales $[\alpha r_{0}/r,r_{0}/r]$, and in particular at scales $[r_{0},1]$.
Applying Proposition 2.1 to this measure and undoing the rescaling, we see
after a routine calculation that

${\operatorname{E}}(\mu|_{I_{i}},r_{0})\leq\varepsilon(r/r_{0})^{4\delta}r_{0}^{\delta}\lesssim
C^{4}\varepsilon r_{0}^{\delta}\mu(2I_{i}-I_{i})^{4}.$

Putting all this together, we see that

${\operatorname{E}}(\mu,r)\lesssim C^{4}\varepsilon
r_{0}^{\delta}\sum_{I_{1},I_{2},I_{3},I_{4}:I_{1}-I_{2}-I_{3}+I_{4}\cap[-4r,4r]\neq\emptyset}\prod_{i=1}^{4}\mu(2I_{i}-I_{i}).$

By the Fubini–Tonelli theorem, we can write the right-hand side as

$C^{4}\varepsilon
r_{0}^{\delta}\int_{\mathbb{R}^{4}}\sum_{I_{1},I_{2},I_{3},I_{4}:I_{1}-I_{2}-I_{3}+I_{4}\cap[-4r,4r]\neq\emptyset}\prod_{i=1}^{4}1_{2I_{i}-I_{i}}(x_{i})\
d\mu(x_{1})\dots d\mu(x_{4}).$

The integrand can be computed to equal $O(1)$, and vanishes unless

$|x_{1}-x_{2}-x_{3}+x_{4}|\leq 100r/r_{0}$

(say). Thus we conclude that

${\operatorname{E}}(\mu,r)\lesssim C^{4}\varepsilon
r_{0}^{\delta}{\operatorname{E}}(\mu,100r/r_{0})$

and the claim now follows by using Lemma 3.1 to remove the factor of $100$. ∎

If we now set $\varepsilon$ to be a small multiple of $1/C^{4}$, we can ensure
that

${\operatorname{E}}(\mu,r)\leq\frac{1}{e}r_{0}^{\delta}{\operatorname{E}}(\mu,r/r_{0})$

and hence upon induction (starting with the base case $r_{0}\leq r\leq 1$) one
has

${\operatorname{E}}(\mu,r)\lesssim(r/r_{0})^{\delta+\frac{1}{\log
r_{0}}}{\operatorname{E}}(\mu,1)$

for all $\alpha\leq r\leq 1$. Substituting the specific value (2.1)
corresponding to the indicated choice of $\varepsilon$, we obtain the claim.

##  4\. Gowers norms and approximate groups

In the previous section we connected the additive energy
${\operatorname{E}}(\mu,r)$ to the Gowers uniformity norms used in additive
combinatorics. We now develop the theory of these norms (and the related
notion of an _approximate group_) in more detail, as we shall rely heavily on
these tools for the higher-dimensional argument. As we will be working in the
continuous setting of Euclidean spaces $\mathbb{R}^{d}$ rather than in the
discrete settings that are more traditional in additive combinatorics, we
shall phrase these concepts in the general setting of locally compact abelian
groups. In this section we will lay out the basic theory of these concepts;
most of the material is standard, except for a key “scale splitting” estimate
which will underlie various manifestations of an “induction on scales”
strategy; see Lemma 4.3 and Lemma 4.8(iii).

######  Definition 4.1 (LCA groups).

An _LCA group_ is a locally compact abelian group $V=(V,+)$ equipped with a
Haar measure $m_{V}$, and the Borel sigma algebra. For any $1\leq
p\leq\infty$, we define the usual Banach spaces $L^{p}(V)$ of
$p^{\mathrm{th}}$ power integrable functions $f\colon V\to\mathbb{C}$,
quotiented out by almost everywhere equivalence. We let $L^{p}(V)_{+}$ denote
the subset of $L^{p}(V)$ consisting of functions that are non-negative (almost
everywhere). For any positive measure subset $X$ of $V$, we define $L^{p}(X)$
and $L^{p}(X)_{+}$ similarly.

In this paper we will mostly be concerned with the case when $V$ is a
Euclidean space $\mathbb{R}^{d}$ equipped with Lebesgue measure, though we
will occasionally also need to work with the lattice $\mathbb{Z}^{d}$ with
counting measure, or hyperplanes
$v^{\perp}\coloneqq\\{x\in\mathbb{R}^{d}:x\cdot v=0\\}$ equipped with Lebesgue
measure.

######  Definition 4.2 (Gowers uniformity norm).

[7, Definition 1.1] Let $V$ be an LCA group. If $f_{1},f_{2},f_{3},f_{4}\in
L^{4/3}(V)$, we define the Gowers inner product

$\langle
f_{1},f_{2},f_{3},f_{4}\rangle_{U^{2}(V)}\coloneqq\int_{V}\int_{V}\int_{V}f_{1}(x)\overline{f_{2}}(x+h)\overline{f_{3}}(x+k)f_{4}(x+h+k)\
dm_{V}(x)dm_{V}(h)dm_{V}(k)$

(these integrals can be shown to be absolutely convergent by the Hölder and
Young inequalities, or by interpolation) and for $f\in L^{4/3}(V)$ we define
the uniformity norm $\|f\|_{U^{2}(V)}=\|f\|_{U^{2}(V,)}$ by the formula

$\|f\|_{U^{2}(V)}\coloneqq\langle f,f,f,f\rangle_{U^{2}(V)}^{1/4}.$

If $\mu$ is an absolutely continuous Radon measure on $V$ whose Radon-Nikodym
derivative $\frac{d\mu}{dm_{V}}$ is finite, we define (by abuse of notation)

$\|\mu\|_{U^{2}(V)}\coloneqq\left\|\frac{d\mu}{dm_{V}}\right\|_{U^{2}(V)}.$

Finally, if $X$ is a positive measure subset of $V$ and $f\in L^{4/3}(X)$, we
define

$\|f\|_{U^{2}(X)}\coloneqq\|f1_{X}\|_{U^{2}(V)}.$

Although we define the Gowers norms here for $f\in L^{4/3}(V)$, for our
applications it would suffice to restrict attention to the non-negative
functions $f\in L^{4/3}(V)_{+}$.

As noted in [7], the norms $\|\|_{U^{2}(V)}$ are indeed norms; in fact one has
an explicit representation

$\|f\|_{U^{2}(V)}=\|f*f\|_{L^{2}(V)}^{1/2}=\|\hat{f}\|_{L^{4}(\hat{V})}$ (4.1)

in terms of the $L^{4}$ norm of the Fourier transform
$\hat{f}\colon\hat{V}\to\mathbb{C}$ on the Pontragyin dual $\hat{V}$ of $V$
(equipped with the dual Haar measure $m_{\hat{V}}$), where we use the usual
convolution operation

$f*g(x)\coloneqq\int_{V}f(x-y)g(y)\ dm_{V}(y).$

Similarly one has

$\langle
f_{1},f_{2},f_{3},f_{4}\rangle_{U^{2}(V)}=\int_{\hat{V}}\hat{f}_{1}(\xi)\overline{\hat{f}_{2}(\xi)}\overline{\hat{f}_{3}(\xi)}\hat{f}_{4}(\xi)\
dm_{\hat{V}}(\xi).$

From Young’s inequality or the Hausdorff-Young inequality one then has the
bound

$\|f\|_{U^{2}(V)}\leq\|f\|_{L^{4/3}(V)}$ (4.2)

so in particular by Hölder’s inequality

$\|f\|_{U^{2}(V)}\leq\|f\|_{L^{1}(V)}^{3/4}\|f\|_{L^{\infty}(V)}^{1/4}$ (4.3)

(compare with (1.2)). From another application of Hölder (or Cauchy-Schwarz)
we conclude the _Gowers–Cauchy–Schwarz inequality_

$|\langle
f_{1},f_{2},f_{3},f_{4}\rangle_{U^{2}(V)}|\leq\|f_{1}\|_{U^{2}(V)}\|f_{2}\|_{U^{2}(V)}\|f_{3}\|_{U^{2}(V)}\|f_{4}\|_{U^{2}(V)}.$
(4.4)

For functions $f\in L^{4/3}(X)$ supported on a set $X$ of positive finite
measure, we also have from Cauchy–Schwarz that

$\|f\|_{U^{2}(X)}=\|f*f\|_{L^{2}(2X)}^{1/2}\geq
m_{V}(2X)^{-1/4}\|f*f\|_{L^{1}(2X)}^{1/2}$

and hence

$\|f\|_{U^{2}(X)}\geq\mu_{V}(2X)^{-1/4}\|f\|_{L^{1}(X)}.$ (4.5)

In view of (4.2), one can think of the $U^{2}(V)$ norm of a function $f$ as
the $L^{4/3}(V)$ norm multiplied by a dimensionless (scale-invariant) quantity
that informally measures the amount of “additive structure” present on $V$.

The Gowers norm is clearly invariant under changes of variable by measure-
preserving affine homomorphisms. In particular it is translation-invariant,
hence by Minkowski’s inequality one has

$\|f*g\|_{U^{2}(V)}\leq\|f\|_{U^{2}(V)}\|g\|_{L^{1}(V)}$

for all $f\in L^{4/3}(V)$ and $g\in L^{1}(V)$. In a similar spirit, one has

$\|f*\mu\|_{U^{2}(V)}\leq\|f\|_{U^{2}(V)}$ (4.6)

for any $f\in L^{4/3}(V)$ and any Radon probability measure $\mu$, where the
convolution is now defined as

$f*\mu(x)\coloneqq\int_{V}f(x-y)\ d\mu(y).$

As mentioned previously, we will primarily be concerned with the Gowers norm
on the non-negative cone $L^{4/3}(V)_{+}$ of $L^{4/3}(V)$. Clearly the Gowers
norm is monotone on this cone in the sense that

$\|f\|_{U^{2}(V)}\leq\|g\|_{U^{2}(V)}$ (4.7)

whenever $f,g\in L^{4/3}(V)_{+}$ are such that $f\leq g$ pointwise almost
everywhere.

If one has two functions $f\in L^{4/3}(V),f^{\prime}\in L^{4/3}(V^{\prime})$
on two LCA groups $V,V^{\prime}$, then we can define the tensor product
$f\otimes f^{\prime}\in L^{4/3}(V\times V^{\prime})$ on the product LCA group
$V\times V^{\prime}$ (equipped with product Haar measure) by the formula

$f\otimes f^{\prime}(x,x^{\prime})\coloneqq f(x)f^{\prime}(x^{\prime})$

for $x\in V$, $x^{\prime}\in V^{\prime}$. From the Fubini–Tonelli theorem we
have the identity

$\|f\otimes f^{\prime}\|_{U^{2}(V\times
V^{\prime})}=\|f\|_{U^{2}(V)}\|f^{\prime}\|_{U^{2}(V^{\prime})}.$ (4.8)

For functions that are not of tensor product form, we have the following
“splitting inequality” that serves as a partial substitute for (4.8):

######  Lemma 4.3 (Splitting inequality).

Let $V,V^{\prime}$ be LCA groups. If $f\in L^{4/3}(V\times V^{\prime})$, then

$\|f\|_{U^{2}(V\times V^{\prime})}\leq\|f_{V}\|_{U^{2}(V^{\prime})}$ (4.9)

where $f_{V}\in L^{4/3}(V^{\prime})$ is the function

$f_{V}(v^{\prime})\coloneqq\|f(\cdot,v^{\prime})\|_{U^{2}(V)}$

where $f(\cdot,v^{\prime})$ is the function $v\mapsto f(v,v^{\prime})$ (this
is well-defined in $L^{4/3}(V)$ for almost every $v^{\prime}\in V^{\prime}$).

The right-hand side of (4.9) can be thought of as an iterated norm
$\|f\|_{U^{2}(V^{\prime};U^{2}(V))}$, so (4.9) can be written as

$\|f\|_{U^{2}(V\times V^{\prime})}\leq\|f\|_{U^{2}(V;U^{2}(V^{\prime}))}$

which can be compared with the Fubini–Tonelli identity

$\|f\|_{L^{4/3}(V\times V^{\prime})}=\|f\|_{L^{4/3}(V;L^{4/3}(V^{\prime}))}.$

Thus one can view Lemma 4.3 as an analogue of the Fubini–Tonelli theorem for
the Gowers uniformity norm $U^{2}$.

###### Proof.

The fact that $f_{V}\in L^{4/3}(V^{\prime})$ follows from (4.2) and the
Fubini–Tonelli theorem. From another application of Fubini–Tonelli one has

$\displaystyle\langle f,f,f,f\rangle_{U^{2}(V\times V^{\prime})}$
$\displaystyle=\int_{V^{\prime}}\int_{V^{\prime}}\int_{V^{\prime}}\langle
f(\cdot,x^{\prime}),f(\cdot,x^{\prime}+h^{\prime}),f(\cdot,x^{\prime}+k^{\prime}),f(\cdot,x^{\prime}+h^{\prime}+k^{\prime})\rangle_{U^{2}(V)}$
$\displaystyle\quad
dm_{V^{\prime}}(x^{\prime})dm_{V^{\prime}}(h^{\prime})dm_{V^{\prime}}(k^{\prime})$

and hence by the Gowers–Cauchy–Schwarz inequality (4.4)

$\langle f,f,f,f\rangle_{U^{2}(V\times
V^{\prime})}\leq\int_{V^{\prime}}\int_{V^{\prime}}\int_{V^{\prime}}f_{V}(x^{\prime})f_{V}(x^{\prime}+h^{\prime})f_{V}(x^{\prime}+k^{\prime})f_{V}(x^{\prime}+h^{\prime}+k^{\prime})dm_{V^{\prime}}(x^{\prime})dm_{V^{\prime}}(h^{\prime})dm_{V^{\prime}}(k^{\prime})$

or equivalently

$\|f\|_{U^{2}(V\times V^{\prime})}^{4}\leq\|f_{V}\|_{U^{2}(V^{\prime})}^{4}$

and the claim follows. ∎

We will be interested in _inverse theorems_ for the trivial inequality (4.3),
that is to say descriptions of those $f$ for which this inequality is close to
sharp. We now recall a key definition (see e.g., [20, Definition 2.25]:

######  Definition 4.4 (Approximate group).

Let $V$ be an LCA group and $K\geq 1$. A subset $H$ of $V$ is said to be a
_$K$ -approximate group_ if $H$ contains the origin, is symmetric (thus
$-H=H$), and if $H+H$ can be covered by at most $K$ translates of $H$. (In
particular, this implies that $m_{V}(mH)\leq K^{m-1}m_{V}(H)$ for all natural
numbers $m$.) If $H$ has positive finite measure, we define the uniform
probability measure $\nu_{H}$ on $H$ by the formula

$\nu_{H}(E)\coloneqq\frac{m_{V}(E\cap H)}{m_{V}(H)}$

for all measurable $E\subset V$. Similarly, if $H$ has finite cardinality, we
define the uniform probability measure $\nu_{H}$ on $H$ by the formula

$\nu_{H}(E)\coloneqq\frac{\\#(E\cap H)}{\\#H}$

(note that these two definitions are compatible with each other when $V$ is
discrete).

######  Examples 4.5.

If $r>0$ and $d\geq 1$, then the ball $B(0,r)=B^{d}(0,r)$ is a
$O(1)^{d}$-approximate group, If $v\in V$ and $M$ is a natural number, then
the arithmetic progression $\\{mv:m=-M,\dots,M\\}$ is an $O(1)$-approximate
group. If $H,H^{\prime}\subset V$ are a $K$-approximate group and
$K^{\prime}$-approximate group respectively, then $H+H^{\prime}$ is a
$KK^{\prime}$-approximate group.

######  Theorem 4.6 (Inverse theorem).

Let $V$ be an LCA group. Let $A,N>0$ and $0<\varepsilon\leq 1/2$, and let
$f\in L^{4/3}(X)_{+}$ for some compact subset $X$ of $V$ obeying the bounds

$\displaystyle\|f\|_{L^{\infty}(X)}$ $\displaystyle\leq A$ (4.10)
$\displaystyle\|f\|_{L^{1}(X)}$ $\displaystyle\leq AN$ (4.11)
$\displaystyle\|f\|_{U^{2}(X)}$ $\displaystyle\geq\varepsilon AN^{3/4}$ (4.12)

(so in particular (4.3) is sharp up to a factor of $\varepsilon$).

  * (i)

(Truncating the small values of $f$) One has

$\|f1_{f\geq\varepsilon^{4}A/16}\|_{U^{2}(X)}\geq\frac{\varepsilon}{2}AN^{1/4}.$

  * (ii)

(Approximate symmetry along an approximate subgroup) There exists a
$\varepsilon^{-O(1)}$-approximate group $H$ in $10X-10X$ of measure
$m_{V}(H)=\varepsilon^{O(1)}N$ such that

$\|f*\nu_{H}\|_{U^{2}(11X-10X)}=\varepsilon^{O(1)}AN^{3/4}.$ (4.13)

Informally, the conclusion of Theorem 4.6 (when compared against (4.12) and
(4.6)) asserts that $f*\nu_{H}$ resembles $f$ in some weak statistical sense,
so that $f$ is “approximately symmetric along $H$”, again in a weak
statistical sense.

###### Proof.

From (4.3), (4.11) one has

$\|f1_{f<\varepsilon^{4}A/16}\|_{U^{2}(X)}\leq\|f\|_{L^{1}(X)}^{3/4}(\varepsilon^{4}A/16)^{1/4}\leq\frac{\varepsilon}{2}AN^{3/4}$

and the claim (i) then follows from (4.12) and the triangle inequality for the
$U^{2}$ norm.

For (ii) our main tool will be the Balog–Szemerédi–Gowers theorem, to conclude
that $f$ looks roughly like the function $A1_{H+y}$ for some translate $H+y$
of an approximate group $H$, which we can then use to establish (4.13).

Let $F\coloneqq\\{x\in X:f(x)\geq\varepsilon^{4}A/16\\}$, then $E\subset X$
and we have the pointwise bound

$f1_{f\geq\varepsilon^{4}A/16}\leq A1_{F}$

and hence by (4.7)

$\|1_{F}\|_{U^{2}(X)}\geq\frac{\varepsilon}{2}N^{3/4}.$

Also, from (4.11) and Markov’s inequality one has

$m_{V}(F)\leq\frac{16}{\varepsilon^{4}}N.$

From (4.3) one has $\|1_{F}\|_{U^{2}(X)}\leq m_{V}(F)^{3/4}$. Comparing these
inequalities we conclude that

$m_{V}(F)=\varepsilon^{O(1)}N$

and

$\|1_{F}\|_{U^{2}(X)}^{4}=\varepsilon^{O(1)}N^{3}.$

In the language of [19, Definition 4.1] (adapted to the additive group $V$),
the quantity $\|1_{F}\|_{U^{2}(X)}^{4}$ is the additive energy
${\operatorname{E}}(F,F)$ of $F$. Applying the Balog–Szemerédi–Gowers theorem
(in the form of [19, Theorem 5.4]), we can then find a
$\varepsilon^{-O(1)}$-approximate group $H$ in $V$ of measure

$m_{V}(H)=\varepsilon^{O(1)}m_{V}(F)=\varepsilon^{O(1)}N$

such that

$m_{V}(F\cap(H+y))=\varepsilon^{O(1)}m_{V}(F)=\varepsilon^{O(1)}N$ (4.14)

for some $y\in V$. An inspection of the proof [19, Theorem 5.4] also reveals
that the approximate group $H$ is constructed to lie in $10F-10F$ (say), and
thus also lies in $10X-10X$; in particular, $f*\nu_{H}$ is supported in
$11X-10X$. From (4.7), (4.14) and the $\varepsilon^{-O(1)}$-approximate group
nature of $H$ we then have

$\displaystyle\|f*\nu_{H}\|_{U^{2}(11X-10X)}$
$\displaystyle\geq\left\|\frac{\varepsilon^{4}A}{16}1_{F\cap(H+y)}*\nu_{H}\right\|_{U^{2}(V)}$
$\displaystyle=\varepsilon^{O(1)}A\|1_{F\cap(H+y)}*\nu_{H}*1_{F\cap(H+y)}*\nu_{H}\|_{L^{2}(V)}^{1/2}$
$\displaystyle=\varepsilon^{O(1)}A\|1_{F\cap(H+y)}*\nu_{H}*1_{F\cap(H+y)}*\nu_{H}\|_{L^{2}(4H+2y)}^{1/2}$
$\displaystyle\geq\varepsilon^{O(1)}Am_{V}(4H+2y)^{-1/4}\|1_{F\cap(H+y)}*\nu_{H}*1_{F\cap(H+y)}*\nu_{H}\|_{L^{1}(4H+2y)}^{1/2}$
$\displaystyle\gtrsim\varepsilon^{O(1)}Am_{V}(H)^{3/4}$
$\displaystyle=\varepsilon^{O(1)}AN^{3/4}$

while from (4.6), (4.3) we have

$\|f*\nu_{H}\|_{U^{2}(11X-10X)}\leq\|f\|_{U^{2}(V)}\leq\|f\|_{L^{1}(V)}^{3/4}\|f\|_{L^{\infty}(V)}^{1/4}\leq
AN^{3/4}$

and the claim (ii) follows. ∎

Approximate groups $H$ (approximately) contain long arithmetic progressions
$\\{mv:m=-M,\dots,M\\}$ for many choices of generator $v$. A precise
formulation of this result is given by the following lemma (cf. [6, Corollary
6.11]):

######  Lemma 4.7 (Approximate groups contain arithmetic progressions).

Let $V$ be an LCA group isomorphic to either a lattice $\mathbb{Z}^{d}$ or a
Euclidean space $\mathbb{R}^{d}$. Let $H$ be a bounded open $K$-approximate
group for some $K\geq 1$. Then for any natural number $M$, the set

$S\coloneqq\\{v\in V:mv\in 8H\forall m=-M,\dots,M\\}$

has measure

$m_{V}(S)\gtrsim\exp\left(-O\left(\log^{O(1)}(2K)\log(2M)\right)\right)m_{V}(H).$

One can also establish this result (with a worse dependence on $M$, and with
$8H$ improved to $4H$) using the Sanders-Croot-Sisask lemma [15], [3], [16].
It is likely that this lemma applies to arbitrary LCA groups, and not just the
lattices and Euclidean spaces, but these are the only cases we will need here.

###### Proof.

First suppose that $V$ is isomorphic to $\mathbb{Z}^{d}$. Applying [16,
Theorem 1.1], we see that $4H$ contains a generalized arithmetic progression
$P$ of dimension $O(\log^{O(1)}(2K))$ and cardinality
$\gtrsim\exp(-O(\log^{O(1)}(2K))\\#H$. By reducing all the lengths of the
progression by $M$, we see that the set

$\\{v\in V:mv\in P\forall m=-M,\dots,M\\}$

has cardinality

$\gtrsim
O(M)^{-O(\log^{O(1)}(2K))}\\#P\gtrsim\exp\left(-O\left(\log^{O(1)}(2K)\log(2M)\right)\right)\\#H.$

As this set is contained in $S$, we obtain the claim (with $8H$ replaced by
$4H$).

For the remaining case we may assume that $V=\mathbb{R}^{d}$ (equipped with
Lebesgue measure). Then for $\varepsilon>0$, $2H\cap\varepsilon\mathbb{Z}^{d}$
is a $O(K^{O(1)})$-approximate group (this follows for instance from [20,
Exercise 2.4.7]), and for $\varepsilon$ small enough the cardinality of this
set is $\sim\varepsilon^{-d}K^{O(1)}m(H)$. Applying the preceding case, we see
that

$\\#S\cap\varepsilon\mathbb{Z}^{d}\gtrsim\exp\left(-O\left(\log^{O(1)}(2K)\log(2M)\right)\right)\varepsilon^{-d}K^{O(1)}m(H).$

Multiplying by $\varepsilon^{d}$ and sending $\varepsilon\to 0$, we obtain the
claim. ∎

Given a Radon measure $\mu$ on an LCA group $V$ and a measurable set $H\subset
V$, we can define the energy

${\operatorname{E}}(\mu,H)\coloneqq\mu^{4}\left(\\{(x_{1},x_{2},x_{3},x_{4})\in(\mathbb{R}^{d})^{4}:x_{1}+x_{2}-x_{3}-x_{4}\in
H\\}\right).$

This generalizes the definition of energy in the introduction; indeed, when
$V=\mathbb{R}^{d}$, we have

${\operatorname{E}}(\mu,r)={\operatorname{E}}(\mu,B(0,r)).$

It is also invariant under affine isomorphisms $T:V\to V^{\prime}$, in the
sense that

${\operatorname{E}}(T_{*}\mu,TH)={\operatorname{E}}(\mu,H)$ (4.15)

for any Radon measure $\mu$ on $V$ and measurable $H\subset V$, where
$T_{*}\mu$ is the pushforward of $\mu$ by $T$.

We now relate these energies to Gowers norms:

######  Lemma 4.8.

Let $V$ be an LCA group, let $\mu$ be a Radon measure on $V$, let $f\in
L^{4/3}(V)_{+}$, and let $H$ be a $K$-approximate group in $V$ of positive
finite measure for some $K\geq 1$.

  * (i)

(Relation between energy and Gowers norm) One has

${\operatorname{E}}(\mu,4H)=K^{O(1)}m_{V}(H)\|\mu*\nu_{H}\|_{U^{2}(V)}^{4}.$

and for any integer $m\geq 4$, one has

${\operatorname{E}}(\mu,mH)=K^{O(m)}{\operatorname{E}}(\mu,4H).$

In particular, for any $m\geq 1$ one has

$\|\mu*\mu_{mH}\|_{U^{2}(V)}=K^{O(m)}\|\mu*\mu_{H}\|_{U^{2}(V)}.$

(Compare with Lemma 3.1.)

  * (ii)

(Shrinking the approximate symmetry group) If $m\geq 1$ is an integer and $P$
is a $K$-approximate group in $mH$ that is either finite or has positive
finite measure, one has

$\|f*\nu_{H}\|_{U^{2}(V)}\leq K^{O(m)}\|f*\nu_{P}\|_{U^{2}(V)}.$

  * (iii)

(Splitting at scale $H$) One has

$\|f\|_{U^{2}(V)}\leq m_{V}(H)^{-3/4}\|f_{4H}\|_{U^{2}(V)}$

where $f_{4H}$ is the local Gowers uniformity norm at scale $H$, defined by

$f_{4H}(y)\coloneqq\|f\|_{U^{2}(4H+y)}.$

Lemma 4.8(iii) asserts, roughly speaking, that to control the “global” $U^{2}$
norm of a function $f$ on an LCA group $V$, it first suffices to control the
“local” $U^{2}$ norm on the “cosets” $4H+y$ of the approximate group $H$, and
then take a further $U^{2}$ norm over the translation parameter $y$ (and apply
a suitable normalization). This fact can be used to give a slightly different
proof of Proposition 3.2 (setting $V=\mathbb{R}$ and $H=[-r,r]$), but we will
not do so here. We will eventually apply Lemma 4.8(iii) to a somewhat unusual
approximate group $H$, namely the sum of a ball and an arithmetic progression.

###### Proof.

From the Fubini–Tonelli theorem, we may expand

$\|\mu*\nu_{H}\|_{U^{2}}^{4}=m_{V}(H)^{-4}\int_{V^{4}}1_{H}*1_{H}*1_{H}*1_{H}(x_{1}+x_{2}-x_{3}-x_{4})\
d\mu(x_{1})\dots d\mu(x_{4}).$

From the pointwise upper bound

$1_{H}*1_{H}*1_{H}*1_{H}\leq m_{V}(H)^{3}1_{4H}$

we conclude that

$\|\mu*\nu_{H}\|_{U^{2}}^{4}\leq m_{V}(H)^{-1}{\operatorname{E}}(\mu,4H).$

Conversely, from the pointwise lower bound

$1_{4H}*1_{4H}*1_{4H}*1_{4H}\geq m_{V}(H)^{3}1_{H}$ (4.16)

we see that

$\|\mu*\nu_{4H}\|_{U^{2}}^{4}\geq m_{V}(H)^{-1}{\operatorname{E}}(\mu,H).$

Replacing $H$ by $mH$ for any natural number $m$ we conclude that

$\|\mu*\nu_{mH}\|_{U^{2}}^{4}\leq m_{V}(H)^{-1}{\operatorname{E}}(\mu,4mH)$

and

${\operatorname{E}}(\mu,mH)\leq
K^{O(m)}m_{V}(H)\|\mu*\nu_{4mH}\|_{U^{2}}^{4}.$

Next, we can upper bound $\nu_{4mH}$ by the sum of at most $K^{O(m)}$
translates of $\nu_{H}$, thus by the triangle inequality

$\|\mu*\nu_{4mH}\|_{U^{2}}^{4}\leq K^{O(m)}\|\mu*\mu_{H}\|_{U^{2}}^{4}.$

Combining these inequalities we obtain (i).

For (ii), we observe the pointwise bound

$\nu_{H}\leq\frac{m_{V}((m+1)H)}{m_{V}(H)}\nu_{(m+1)H}*\nu_{P}$

and thus by (4.6)

$\|f*\nu_{H}\|_{U^{2}(V)}\leq\frac{m_{V}((m+1)H)}{m_{V}(H)}\|f*\nu_{P}\|_{U^{2}(V)}.$

The claim (ii) now follows from the $K$-approximate group nature of $H$.

Now we prove (iii). From (4.16) we have

$\|1_{4H}\|_{U^{2}(V)}\geq m_{V}(H)^{3/4}$

and hence by (4.8) we have

$\|f\otimes 1_{4H}\|_{U^{2}(V\times V)}\geq m_{V}(H)^{3/4}\|f\|_{U^{2}(V)}.$

Applying the measure-preserving transformation $(x,h)\mapsto(x+h,h)$ on
$V\times V$ we conclude that

$\|F\|_{U^{2}(V\times V)}\geq m_{V}(H)^{3/4}\|f\|_{U^{2}(V)}$

where $F(x,h)\coloneqq f(x+h)1_{4H}(h)$. Applying Lemma 4.3, we have

$\|F\|_{U^{2}(V\times V)}\leq\|f_{4H}\|_{U^{2}(V)},$

and the claim (iii) follows. ∎

##  5\. The higher dimensional case

We are now ready to establish Theorem 1.5. By repeating the arguments in
Section 3 (which extend to higher dimensions without difficulty), it suffices
to establish the following proposition.

######  Proposition 5.1 (Slight gain over the trivial bound).

Let $d\geq 1$ be an integer, let $0<\delta<d$ be a non-integer, and let $C>1$
and $0<\varepsilon\leq 1/2$. Let $0<r_{0}<1$ be sufficiently small depending
on $d,\delta,C,\varepsilon$. Let $X\subset\mathbb{R}^{d}$ be a
$\delta$-regular set on scales $[r_{0},1]$ with constant $C$, and let
$\mu_{X}$ be an associated regular measure. Then we have

${\operatorname{E}}(\mu|_{B(0,1)},r_{0})\leq\varepsilon r_{0}^{\delta}.$

In fact one can take to be quasipolynomial in $C/\varepsilon$, in the sense
that

$r_{0}=\exp\left(-\exp\left(O_{\delta,d}\left(\log^{O_{\delta,d}}(C/\varepsilon)\right)\right)\right).$
(5.1)

To prove this proposition we will use induction on the ambient dimension $d$.
To facilitate the induction, it is convenient to relax the hypotheses on $\mu$
somewhat. More precisely, our induction hypothesis will be as follows.

######  Proposition 5.2 (Induction hypothesis).

Let Let $d\geq 1$ be an integer, let $0<\delta<d$ be a non-integer, and let
$C>1$ and $0<\varepsilon\leq 1/2$. Let $r_{0}$ be the quantity (5.1). Let
$X\subset B(0,1)$ be a compact set with the following property:

  * (i)

(Upper $\delta$-regularity) For any $r_{0}\leq r_{2}\leq r_{1}\leq 1$ and any
$x\in\mathbb{R}^{d}$, the set $X\cap B(x,r_{1})$ can be covered by at most
$C(r_{1}/r_{2})^{\delta}$ balls of radius $r_{2}$.

Let $\mu$ be a Radon measure supported on $X$ obeying the upper regularity
bound

$\mu(B(x,r_{1}))\leq Cr_{1}^{\delta}$ (5.2)

for all $r_{0}\leq r_{1}\leq 1$. Then

${\operatorname{E}}(\mu,r_{0})\leq\varepsilon r_{0}^{\delta}.$

The main advantage of working with Proposition 5.2 instead of with Proposition
5.1 is that all the hypotheses on $\mu$ and $X$ are of “upper bound” type
rather than “lower bound” type, so that it becomes easier to remove unwanted
portions of $\mu$ or $X$ as needed.

A simple covering argument shows that if $X$ is a $\delta$-regular set on
scales $[r_{0},1]$ then the property (i) of Proposition 5.2 is satisfied with
$C$ replaced by $O_{\delta,d,C}(1)$, and the property (5.2) is immediate from
the definition of a $\delta$-regular measure. Hence the general dimension case
of Proposition 5.1 follows from Proposition 5.2.

To prove Proposition 5.2, we induct on $d$, assuming that the claim has
already been proven for dimension $d-1$ (this induction hypothesis is vacuous
when $d=1$). Let $0<\delta<d$ be non-integer, let $C>1$ and $\varepsilon>0$,
and let $r_{0}>0$ be chosen later (eventually it will be of the form (5.1)).
Let $X,\mu_{X}$ obey the hypotheses of Proposition 5.2. We assume for sake of
contradiction that

${\operatorname{E}}(\mu,r_{0})>\varepsilon r_{0}^{\delta}.$ (5.3)

To abbreviate notation we now allow all implied constants in asymptotic
notation to depend on $\delta,d$. From Lemma 4.8(i) we conclude that

$\|\mu*\nu_{B(0,r_{0})}\|_{U^{2}(B(0,2))}\gtrsim\varepsilon^{O(1)}r_{0}^{(\delta-d)/4}.$

Informally, this estimate asserts that $X+B(0,r_{0})$ has high additive
energy. We now use the inverse theory to obtain some regularity along a non-
trivial arithmetic progression $P$. From (5.2) we also have the uniform bound

$\|\mu*\nu_{B(0,r_{0})}\|_{L^{\infty}(B(0,2))}\lesssim Cr_{0}^{\delta-d}$

and from (5.2) (with $r=1$) and Young’s inequality one has

$\|\mu*\nu_{B(0,r_{0})}\|_{L^{1}(B(0,2))}\lesssim C.$

We can now apply Theorem 4.6(ii) to conclude that there exists a
$(C/\varepsilon)^{O(1)}$-approximate group $H$ in $B(0,20)$ of measure
$|H|=(C/\varepsilon)^{O(1)}r_{0}^{d-\delta}$ such that

$\|\mu*\nu_{B(0,r_{0})}*\nu_{H}\|_{U^{2}(B(0,20))}=(C/\varepsilon)^{O(1)}r_{0}^{(\delta-d)/4}.$
(5.4)

Informally, this estimate asserts that $X+B(0,r_{0})$ not only has high
additive energy, but also behaves like a union of translates of
$B(0,r_{0})+H$.

Let $M$ be a large integer (depending on $C,d,\delta,\varepsilon$) to be
chosen later. By Lemma 4.7, the set

$S\coloneqq\\{v\in\mathbb{R}^{d}:mv\in 8H\forall m=-10^{3}M,\dots,10^{3}M\\}$

has measure

$|S|\gtrsim M^{-O(\log^{O(1)}(C/\varepsilon))}|H|\sim
M^{-O(\log^{O(1)}(C/\varepsilon))}r_{0}^{d-\delta}.$

In particular, since $\delta>0$, we have $|S|>|B(0,r_{0})|$ if $r_{0}$ is
sufficiently small, and more specifically we can take $r_{0}$ of the form

$r_{0}=M^{-O(\log^{O(1)}(C/\varepsilon))}.$ (5.5)

We conclude that there is a vector $v\in\mathbb{R}^{d}$ with $|v|>r_{0}$ such
that the progression

$P\coloneqq\\{mv:m=-M,\dots,M\\}$

is such that $10^{3}P\subset 8H$. In particular we must have the scale
relations

$r_{0}\leq|v|\leq Mv\leq 1.$

From (5.4) and Lemma 4.8(ii) we conclude that

$\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(0,20))}\gtrsim(C/\varepsilon)^{O(1)}r_{0}^{(\delta-d)/4}.$
(5.6)

Informally, this estimate asserts that $X+B(0,r_{0})$ not only has high
additive energy, but also behaves like a union of translates of the set
$B(0,r_{0})+P$, which is an arithmetic progression of small balls.

Now that we have obtained some regularity along a progression $P$, the next
step is to localize at the scale $M|v|$ of the diameter of $P$. For any
$y\in\mathbb{R}^{d}$, the quantity

$\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y,M|v|))}$ (5.7)

vanishes unless $y$ lies within $O(M|v|)$ of $X$, which constraints $y$ to a
set of measure $O((M|v|)^{d-\delta})$ thanks to the hypothesis (i). From (4.3)
we conclude that the $U^{2}$ norm of the quantity (5.7) (viewed as a function
of $y$) is at most

$\lesssim(M|v|)^{3(d-\delta)/4}\sup_{y\in\mathbb{R}^{d}}\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y,M|v|))}$

and hence by Lemma 4.8(iii) we have

$\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(0,20))}\lesssim(C/\varepsilon)^{O(1)}(M|v|)^{-3\delta/4}\sup_{y\in\mathbb{R}^{d}}\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y,M|v|))}.$

Comparing this with (5.6), we conclude that there exists
$y_{0}\in\mathbb{R}^{d}$ such that

$\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y_{0},M|v|))}\gtrsim(C/\varepsilon)^{O(1)}(M|v|)^{3\delta/4}r_{0}^{(\delta-d)/4}.$

Informally, this estimate asserts that $(X+B(0,r_{0}))\cap B(y_{0},M|v|)$ not
only has high additive energy, but also behaves like a union of translates of
the set $B(0,r_{0})+P$, which has diameter comparable to that of the ball
$B(y_{0},M|v|)$.

Fix this $y_{0}$. We now apply Lemma 4.8(iii) using the $O(1)$-approximate
group

$H\coloneqq B(0,|v|)+P$

(which geometrically is approximately a cylinder of dimensions $|v|\times
M|v|$, oriented in the direction of $v$) to conclude that

$\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y_{0},M|v|))}\lesssim(C/\varepsilon)^{O(1)}(M|v|^{d})^{-3/4}\|f\|_{U^{2}(\mathbb{R}^{d})}$

where $f(y)$ is the local Gowers norm

$f(y)\coloneqq\|\mu*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(B(y_{0},M|v|)\cap(B(y,4|v|)+4P)}.$

The function $f$ vanishes unless $y\in B(y_{0},10M|v|)$, thus

$\|f\|_{U^{2}(B(y_{0},10M|v|))}\gtrsim(C/\varepsilon)^{O(1)}M^{3(\delta+1)/4}|v|^{3(\delta+d)/4}r_{0}^{(\delta-d)/4}.$
(5.8)

Informally, this estimate asserts that the collection of cosets $H+y$ of the
“cylinder” $H$ in which $(X+B(0,r_{0}))\cap B(y_{0},M|v|)$ has high additive
energy, itself has high additive energy.

We now study the quantity $f(y)$ for some $y\in(y_{0},10M|v|)$. We can use
monotonicity and the triangle inequality to bound

$\displaystyle f(y)$
$\displaystyle\leq\|\mu|_{B(y,5|v|)+5P}*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq\sum_{z\in
y+5P}\|\mu|_{B(z,5|v|)}*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(\mathbb{R}^{d})}.$

The summand vanishes unless $z$ lies in $X+B(0,5|v|)$. We also have from (5.2)
and Young’s inequality that

$\|\mu|_{B(z,5|v|)}*\nu_{B(0,r_{0})}*\nu_{P}\|_{L^{1}(\mathbb{R}^{d})}\leq\mu(B(z,5|v|))\lesssim(C/\varepsilon)^{O(1)}|v|^{\delta}$

and

$\|\mu|_{B(z,5|v|)}*\nu_{B(0,r_{0})}*\nu_{P}\|_{L^{\infty}(\mathbb{R}^{d})}\lesssim(C/\varepsilon)^{O(1)}M^{-1}r_{0}^{\delta-d}$

and hence by (4.3) we have

$\|\mu|_{B(z,5|v|)}*\nu_{B(0,r_{0})}*\nu_{P}\|_{U^{2}(\mathbb{R}^{d})}\lesssim(C/\varepsilon)^{O(1)}M^{-1/4}r_{0}^{(\delta-d)/4}|v|^{3\delta/4}.$

We thus have

$f(y)\lesssim(C/\varepsilon)^{O(1)}M^{-1/4}r_{0}^{(\delta-d)/4}|v|^{3\delta/4}\\#((y+5P)\cap(X+B(0,5|v|))).$

If we introduce the function

$F(y)\coloneqq 1_{B(y_{0},10M|v|)\cap(X+B(0,10|v|))}$

then a simple volume-packing argument shows that

$\\#((y+5P)\cap(X+B(0,5|v|)))\lesssim(C/\varepsilon)^{O(1)}MF*\nu_{B(0,10|v|)+10P}(y)$

and thus we have the pointwise bound

$f\lesssim(C/\varepsilon)^{O(1)}M^{3/4}r_{0}^{(\delta-d)/4}|v|^{3\delta/4}F*\nu_{B(0,10|v|)+10P}.$

Inserting this into (5.8), we conclude that

$\|F*\nu_{B(0,10|v|)+10P}\|_{U^{2}(\mathbb{R}^{d})}\gtrsim(C/\varepsilon)^{O(1)}M^{3\delta/4}|v|^{3d/4}.$

Clearly we have

$\|F*\nu_{B(0,10|v|)+10P}\|_{L^{\infty}(\mathbb{R}^{d})}\leq\|F\|_{L^{\infty}(\mathbb{R}^{d})}\leq
1$

while from the hypothesis (i) the support of $F$ is covered by $O(M^{\delta})$
balls of radius $|v|$, and hence

$\|F*\nu_{B(0,10|v|)+10P}\|_{L^{1}(\mathbb{R}^{d})}\leq\|F\|_{L^{1}(\mathbb{R}^{d})}\lesssim
M^{\delta}|v|^{d}.$ (5.9)

Applying Theorem 4.6(i), we conclude that

$\|(F*\nu_{B(0,10|v|)+10P})1_{G}\|_{U^{2}(\mathbb{R}^{d})}\gtrsim(C/\varepsilon)^{O(1)}M^{3\delta/4}|v|^{3d/4}$
(5.10)

where $G$ is a set of the form

$G\coloneqq\\{y:F*\nu_{B(0,10|v|)+10P}(y)\geq(C/\varepsilon)^{-C_{0}}\\}$

for some sufficiently large $C_{0}$ depending only on $d,\delta$. Note that
$G$ is a compact subset of $B(y_{0},100M|v|)$. From (5.9), (5.10) we conclude
that

$\|1_{G}\|_{U^{2}(B(y_{0},100M|v|))}\gtrsim(C/\varepsilon)^{O(1)}M^{3\delta/4}|v|^{3d/4}.$
(5.11)

Informally, this estimate asserts that the collection of cosets $H+y$ of the
“cylinder” $H$ in which $(X+B(0,r_{0}))\cap B(y_{0},M|v|)$ has large density,
itself has high additive energy.

We now split into two cases: the low-dimensional case $\delta<1$ and the high-
dimensional case $\delta>1$ (recall that $\delta$ is assumed to be non-
integer). In the low-dimensional case we observe that for any
$y\in\mathbb{R}^{d}$, the only portion of $X$ that contributes to
$F*\nu_{B(0,10|v|)+10P}(y)$ lies in $B(y,100M|v|)$, and is thus covered by
$O(M^{\delta})$ balls of radius $|v|$ thanks to the hypothesis (i). This leads
to the pointwise estimate

$F*\nu_{B(0,10|v|)+10P}(y)\lesssim(C/\varepsilon)^{O(1)}M^{-1}M^{\delta};$

as we are in the low-dimensional case $\delta<1$, taking
$M=(C/\varepsilon)^{C_{1}}$ for a sufficiently large $C_{1}$ (depending on
$d,\delta$) sufficiently large will then imply that the set $G$ is empty,
which contradicts (5.11), with $r_{0}$ of the required size thanks to (5.5).
(Informally, the point is that the set $X$ is too low dimensional to
adequately fill out a coset $H+y$.)

Now suppose we are in the high-dimensional case $\delta>1$, which of course
forces $d\geq 2$. Here we shall “quotient out” by $P$ and use the induction
hypothesis.

Let $v^{\perp}\coloneqq\\{x\in\mathbb{R}^{d}:x\cdot v=0\\}$ be the hyperplane
in $\mathbb{R}^{d}$ orthogonal to $v$, and let $\pi\colon\mathbb{R}^{d}\to
v^{\perp}$ be the orthogonal projection.

As $G$ is a compact subset of $B(y_{0},100M|v|)$, $\pi(G)$ is a compact subset
of $B_{v^{\perp}}(\pi(y_{0}),100M|v|)$, where we use $B_{v^{\perp}}$ to denote
the balls in the hyperplane $v^{\perp}$. After applying a rigid motion to
identify $v^{\perp}$ with $\mathbb{R}^{d-1}$, one can view $G$ as a subset of
$\pi(G)\times[-100M|v|,100M|v|]$, hence by (4.7), (4.8), (4.3) we have

$\|1_{G}\|_{U^{2}(B(y_{0},100M|v|))}\lesssim(C/\varepsilon)^{O(1)}(M|v|)^{3/4}\|1_{\pi(G)}\|_{U^{2}(B_{v^{\perp}}(\pi(y_{0}),100M|v|))}$

and thus by (5.11)

$\|1_{\pi(G)}\|_{U^{2}(B_{v^{\perp}}(\pi(y_{0}),100M|v|))}\gtrsim(C/\varepsilon)^{O(1)}M^{3(\delta-1)/4}|v|^{3(d-1)/4}.$
(5.12)

Informally, this estimate asserts that the set $G$ (which roughly speaking
tracked the translates of $H+y$ in which $(X+B(0,r_{0}))\cap B(y_{0},M|v|)$
had large density) continues to have large additive energy after applyign the
orthogonal projection $\pi$.

The set $\pi(G)$ also has “dimension $\delta-1$” in the scale range
$[|v|,M|v|]$ in the following sense:

######  Lemma 5.3 ($\pi(G)$ is $\delta-1$-dimensional).

For any $|v|\leq r_{2}\leq r_{1}\leq M|v|$ and any $x\in v^{\perp}$, the set
$\pi(G)\cap B_{v^{\perp}}(x,r_{1})$ can be covered by at most
$(C/\varepsilon)^{O(1)}(r_{1}/r_{2})^{\delta-1}$ balls of radius $r_{2}$ in
$v^{\perp}$.

###### Proof.

Let $\Sigma$ be any $100r_{2}$-separated subset of $\pi(G)\cap
B_{v^{\perp}}(x,r_{1})$. It will suffice to show that
$\\#\Sigma\lesssim(C/\varepsilon)^{O(1)}(r_{1}/r_{2})^{\delta-1}$. Each
$\sigma\in\Sigma$ can be written as $\pi(y_{\sigma})$ for some $y_{\sigma}\in
G$.

By construction of $G$ and $F$, we have

$|(B(y_{\sigma},10|v|)+10P)\cap(X+B(0,20|v|))|\gtrsim(C/\varepsilon)^{O(1)}M|v|^{d}$

for all $\sigma\in\Sigma$. In particular, it requires at least
$\gtrsim(C/\varepsilon)^{O(1)}M|v|/r_{2}$ balls of radius $r_{2}$ to cover
this set. As these sets are at least $10r_{2}$-separated (say) from each other
as $\sigma$ varies, we conclude that the set

$\bigcup_{\sigma\in\Sigma}(B(y_{\sigma},10|v|)+10P)\cap(X+B(0,20|v|))$

requires at least $\gtrsim(C/\varepsilon)^{O(1)}\frac{M|v|}{r_{2}}\\#\Sigma$
balls of radius $r_{2}$ to cover this set. On the other hand, this set is
contained in

$(X+B(0,20|v|))\cap B(y_{0},200M|v|)\cap\pi^{-1}(B_{v^{\perp}}(x,20r_{1})).$
(5.13)

The set $B(y_{0},200M|v|)\cap\pi^{-1}(B_{v^{\perp}}(x,20r_{1}))$ (which is
shaped roughly like a $r_{1}\times M|v|$ cylinder) can be covered by
$O(M|v|/r_{1})$ balls of radius $r_{1}$. Applying hypothesis (i), we conclude
that the set (5.13) can be covered by
$O((C/\varepsilon)^{O(1)}(M|v|/r_{1})(r_{1}/r_{2})^{\delta})$ balls of radius
$r_{2}$. Comparing these bounds, we obtain the claim. ∎

Let $\mu^{\prime}$ denote the measure

$\mu^{\prime}\coloneqq|v|^{1-d}M^{1-\delta}m_{v^{\perp}}|_{\pi(G)+B_{v^{\perp}}(0,|v|)}$

where $m_{v^{\perp}}$ is Lebesgue measure on $v^{\perp}$. From Lemma 5.3
(taking $r_{2}=|v|$) we see that

$\mu^{\prime}(B_{v^{\perp}}(y,r))\lesssim(C/\varepsilon)^{O(1)}(r/M|v|)^{\delta-1}$

for all $r\in[|v|,M|v|]$. Applying the induction hypothesis Proposition 5.2
(with $d$ replaced by $d-1$, $\delta$ replaced by $\delta-1$, and $r_{0}$
replaced by $1/M$) and an affine change of variable to rescale $v^{\perp}$ to
$\mathbb{R}^{d-1}$ and $B_{v^{\perp}}(y_{0},100M|v|)$ to $B^{d-1}(0,1)$, we
conclude from (4.15) that

${\operatorname{E}}(\mu^{\prime},|v|)\leq\varepsilon^{\prime}M^{-(\delta-1)}$

for any $\varepsilon^{\prime}>0$, if $M$ is sufficiently large depending on
$C,\delta,d,\varepsilon,\varepsilon^{\prime}$; indeed we can take $M$ to be of
the form

$M=\exp\left(\exp(O(\log^{O(1)}(C/\varepsilon\varepsilon^{\prime})))\right).$

Applying Lemma 4.8(i), we conclude that

$\|\mu^{\prime}*\nu_{B_{v^{\perp}}(0,|v|)}\|_{U^{2}(v^{\perp})}\lesssim(C/\varepsilon)^{O(1)}(\varepsilon^{\prime})^{1/4}|v|^{-(d-1)/4}M^{-(\delta-1)/4}.$

Using the pointwise bound

$\mu^{\prime}*\nu_{B_{v^{\perp}}(0,|v|)}\gtrsim(C/\varepsilon)^{O(1)}|v|^{1-d}M^{1-\delta}1_{\pi(G)}$

we conclude that

$\|1_{\pi(G)}\|_{U^{2}(v^{\perp})}\lesssim(C/\varepsilon)^{O(1)}(\varepsilon^{\prime})^{1/4}|v|^{3(d-1)/4}M^{3(\delta-1)/4}.$

By taking $\varepsilon^{\prime}$ to equal $(\varepsilon/C)^{C_{2}}$ for a
sufficiently large $C_{2}$ depending only on $d,\delta$, we contradict (5.12),
giving the claim, with the right size (5.1) for $r_{0}$ thanks to (5.5) and
the choice of parameters $\varepsilon^{\prime},M$.

##  6\. Nonlinear expansion

In this section we establish Theorem 1.7. We establish the higher dimensional
case $d>1$ here; the one-dimensional case $d=1$ is proven similarly. The
strategy is to localize to a small enough scale that the nonlinear function
$F$ can be well approximated by a linear one, at which point one can apply the
Cauchy–Schwarz inequality to control the relevant expressions by additive
energies, so that Theorems 1.4, 1.5 may be applied.

We allow all constants to depend on $\delta,d,F$. Let $\varepsilon>0$ be a
small quantity depending on $\delta,d,F$ to be chosen later; by shrinking $r$
as necessary we may assume that $r$ is sufficiently small depending on
$\varepsilon,\delta,d,F$. Let $y_{0}$ be a point in $Y$, thus $y_{0}\in
B(0,2)$. We will only work in the $\varepsilon r^{1/2}$-neighborhood of
$y_{0}$ in $Y_{r}$. Indeed, it will suffice to establish the bound

$|F(X_{r},Y_{r}\cap B(y_{0},\varepsilon
r^{1/2}))|\gtrsim_{C,\varepsilon}r^{d-\delta-\beta}.$

By regularity, we can cover $X_{r}$ by $O_{\varepsilon}(r^{-\delta/2})$ balls
$B(x_{i},\varepsilon r^{1/2})$ of radius $\varepsilon r^{1/2}$ with $x_{i}\in
X\cap B(0,2)$. Using Taylor expansion, the $C^{2}$ nature of $F$ and the non-
vanishing of the differential map $D_{x}F(x,y)$, we see that any two sets

$F(B(x_{i},\varepsilon r^{1/2}),B(y_{0},\varepsilon
r^{1/2})),F(B(x_{j},\varepsilon r^{1/2}),B(y_{0},\varepsilon r^{1/2}))$

with $|x_{i}-x_{j}|\leq\varepsilon$ will be disjoint unless
$|x_{i}-x_{j}|\lesssim\varepsilon r^{1/2}$. From this we conclude that the
sets $F(B(x_{i},\varepsilon r^{1/2}),B(y_{0},\varepsilon r^{1/2}))$ have
overlap $O_{\varepsilon}(1)$, so it will suffice to establish the bound

$|F(X_{r}\cap B(x_{0},\varepsilon r^{1/2}),Y_{r}\cap B(y_{0},\varepsilon
r^{1/2}))|\gtrsim_{C,\varepsilon}r^{d-\delta/2-\beta}$

for each $x_{0}\in X\cap B(0,2)$.

By Taylor expansion and the $C^{2}$ nature of $F$, for $x\in
B(x_{0},\varepsilon r^{1/2})$ and $y\in B(y_{0},\varepsilon r^{1/2})$ one has

$F(x,y)=F(x_{0},y_{0})+D_{x}F(x_{0},y_{0})(x-x_{0})+D_{y}F(x_{0},y_{0})(y-y_{0})+O(\varepsilon
r).$

From this and the invertibility of $D_{x}F(x_{0},y_{0}),D_{y}F(x_{0},y_{0})$
we see that the set

$F(X_{r}\cap B(x_{0},\varepsilon r^{1/2}),Y_{r}\cap B(y_{0},\varepsilon
r^{1/2}))$

contains the set

$F(x_{0},y_{0})+D_{x}F(x_{0},y_{0})((X_{r/2}\cap B(x_{0},\varepsilon
r^{1/2}))-x_{0})+D_{y}F(x_{0},y_{0})((Y_{r/2}\cap B(y_{0},\varepsilon
r^{1/2}))-y_{0})$

so after subtracting a constant and inverting the linear transformation
$D_{x}F(x_{0},y_{0})$ it will suffice to establish the bound

$|A+B|\gtrsim_{C,\varepsilon}r^{d-\delta/2-\beta}$ (6.1)

where

$A\coloneqq X_{r/2}\cap B(x_{0},\varepsilon r^{1/2})$

and

$B\coloneqq D_{x}F(x_{0},y_{0})^{-1}D_{y}F(x_{0},y_{0})(Y_{r/2}\cap
B(y_{0},\varepsilon r^{1/2})).$

From the regular nature of $X$ one sees from Definition 1.1 and standard
volume-packing calculations that

$|A|\sim_{C,\varepsilon}r^{d-\delta/2}$

and a similar argument (using also the invertibility of $D_{y}F$ and the
$C^{2}$ nature of $F$) gives

$|B|\sim_{C,\varepsilon}r^{d-\delta/2}.$ (6.2)

In particular

$\|1_{A}*1_{B}\|_{L^{1}(\mathbb{R}^{d})}\sim_{C,\varepsilon}r^{2(d-\delta/2)}.$

Since $1_{A}*1_{B}$ is supported on $A+B$, to prove (6.1) it thus suffices by
Cauchy-Schwarz to show that

$\|1_{A}*1_{B}\|_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim_{C,\varepsilon}r^{3(d-\delta/2)+\beta}.$

By the Gowers-Cauchy-Schwarz inequality we have

$\|1_{A}*1_{B}\|_{L^{2}(\mathbb{R}^{d})}^{2}\leq\|1_{A}\|_{U^{2}(\mathbb{R}^{d})}^{2}\|1_{B}\|_{U^{2}(\mathbb{R}^{d})}^{2}.$

From (4.2), (6.2) we have

$\|1_{B}\|_{U^{2}(\mathbb{R}^{d})}^{4}\lesssim_{C,\varepsilon}r^{3(d-\delta/2)}$

so it will suffice to establish the bound

$\|1_{A}\|_{U^{2}(\mathbb{R}^{d})}^{4}\lesssim_{C,\varepsilon}r^{3(d-\delta/2)+\beta}.$

From the pointwise estimate

$1_{A}\lesssim 1_{X_{r}\cap B(x_{0},r^{1/2})}*\nu_{B(0,r)}$

and Lemma 4.8(i) (or the higher-dimensional version of Lemma 3.1) it suffices
to show that

${\operatorname{E}}(1_{X_{r}\cap
B(x_{0},r^{1/2})}dm,r)\lesssim_{C,\varepsilon}r^{4d-3\delta/2+\beta}$

(where $dm$ is Lebesgue measure). We rescale this as

${\operatorname{E}}(r^{(\delta-d)/2}1_{\tilde{X}_{r^{1/2}}\cap
B(0,1)}dm,r)\lesssim_{C,\varepsilon}r^{\delta/2+\beta}$

where $\tilde{X}$ is a rescaled version of $X$:

$\tilde{X}\coloneqq\left\\{\frac{x-x_{0}}{r^{1/2}}:x\in X\right\\}.$

From the regularity of $X$ and a routine change of variables we check that
$\tilde{X}$ is $\delta$-regular at scales $[r^{1/2},1]$ (with constant
$C^{O(1)}$), and that $r^{(\delta-d)/2}1_{\tilde{X}_{r^{1/2}}\cap B(0,1)}dm$
is an associated regular measure (again with constant $C^{O(1)}$). The claim
now follows from Theorem 1.5 (adjusting the constants in the definition of
$\beta$ appropriately).

######  Remark 6.1.

The regularity hypotheses on $Y$ can be relaxed substantially; in fact with a
little more effort one could replace $Y_{r}$ here by any subset of $B(0,1)$ of
measure $\gtrsim_{C,\delta}r^{d-\delta}$. We leave the details to the
interested reader.

##  7\. From additive energy to the fractal uncertainty principle

We now prove Theorem 1.8. Let the notation and hypotheses be as in that
theorem. Clearly we have

$\|{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{1}(\mathbb{R}^{d})\to
L^{\infty}(\mathbb{R}^{d})}\lesssim_{d}h^{-d/2}$

so by the Riesz–Thorin theorem it suffices to show that

$\|{\mathcal{F}}_{h}1_{Y_{h}}\|_{L^{\infty}(\mathbb{R}^{d})\to
L^{4}(\mathbb{R}^{d})}\lesssim_{d}h^{d/2-3\delta/4+2\beta}.$

Let $f\in L^{\infty}(\mathbb{R})$ be of norm one. From (4.1) and a rescaling
we have

$\|{\mathcal{F}}_{h}(f1_{Y_{h}})\|_{L^{4}(\mathbb{R}^{d})}\sim_{d}h^{d/4}\|f1_{Y_{h}}\|_{U^{2}(\mathbb{R}^{d})}.$

As $Y$ is $\delta$-regular, we have the pointwise bound

$f1_{Y_{h}}\lesssim_{d}Ch^{d-\delta}\mu_{Y}*\nu_{B(0,2h)}$

and hence

$\|{\mathcal{F}}_{h}(f1_{Y_{h}})\|_{L^{4}(\mathbb{R}^{d})}\lesssim_{d}Ch^{3d/4-\delta}\|\mu_{Y}*\nu_{B(0,2h)}\|_{U^{2}(\mathbb{R}^{d})}.$

Applying Lemma 4.8(i) we conclude that

$\|{\mathcal{F}}_{h}(f1_{Y_{h}})\|_{L^{4}(\mathbb{R}^{d})}\lesssim_{d}Ch^{d/2-\delta}{\operatorname{E}}(\mu_{Y},\delta)^{1/4}$

and the claim (1.13) now follows from Theorems 1.4, 1.5, after adjusting
$\beta$ as necessary. The claim (1.14) then follows from Hölder’s inequality
after observing from the $\delta^{\prime}$-regularity of $X$ that

$|X_{h}|\lesssim_{C^{\prime},\delta^{\prime},d}h^{d-\delta^{\prime}}.$

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