# A Marstrand projection theorem for lines

Shengwen Gan  Department of Mathematics
University of Wisconsin-Madison
Madison, WI-53706, USA<EMAIL_ADDRESS>
###### Abstract.

Fix integers $1<k<n$. For $V\in G(k,n)$, let $P_{V}:\mathbb{R}^{n}\rightarrow
V$ be the orthogonal projection. For $V\in G(k,n)$, define the map

$\pi_{V}:A(1,n)\rightarrow A(1,V)\bigsqcup V.$ $\ell\mapsto P_{V}(\ell).$

For any $0<a<\textup{dim}(A(1,n))$, we find the optimal number $s(a)$ such
that the following is true. For any Borel set $\mathbf{A}\subset A(1,n)$ with
$\dim(\mathbf{A})=a$, we have

$\dim(\pi_{V}(\mathbf{A}))=s(a),\textup{~{}for~{}a.e.~{}}V\in G(k,n).$

When $A(1,n)$ is replaced by $A(0,n)=\mathbb{R}^{n}$, it is the classical
Marstrand projection theorem, for which $s(a)=\min\\{k,a\\}$. A new ingredient
of the paper is the Fourier transform on affine Grassmannian.

###### Key words and phrases:

Marstrand projection theorem, exceptional set estimate, Fourier analysis,
Brascamp-Lieb inequality

###### 2020 Mathematics Subject Classification:

28A75, 28A78

##  1\. Introduction

For $k<n$, let $G(k,n)$ be the set of $k$-dimensional subspaces in
$\mathbb{R}^{n}$, and $A(k,n)$ be the set of $k$-dimensional affine spaces in
$\mathbb{R}^{n}$. For $V\in G(k,n)$ and $l<k$, let $G(l,V)$ be the set of
$l$-dimensional subspaces in $V$, and $A(l,V)$ be the set of $l$-dimensional
affine spaces in $V$. For simplicity, we just call $l$-dimensional affine
space as $l$-plane. Fix integers $1<k<n$. For $V\in G(k,n)$, let

$P_{V}:\mathbb{R}^{n}\rightarrow V$

be the orthogonal projection onto $V$.

Let $V\in G(k,n)$. Note that for $L\in A(1,n)$, we have $P_{V}(L)$ is either a
line or a point in $V$. We can define

$\pi_{V}:A(1,n)\rightarrow A(1,V)\bigsqcup V$ $L\mapsto P_{V}(L).$

Marstrand [8] proved the following result. Let $A\subset\mathbb{R}^{n}$ with
$\dim(A)=a$. Then

(1) $\dim(P_{V}(A))=\min\\{a,k\\},\textup{~{}for~{}a.e.~{}}V\in G(k,n).$

Naturally, one can consider the following Marstrand-type projection problem
for $A(1,n)$.

For $0<a<2(n-1)=\dim(A(1,n))$, what is the optimal number $s(a)$ such that the
following is true? Let $\mathbf{A}\subset A(1,n)$ with $\dim(\mathbf{A})=a$.
(Since $A(1,n)$ is a Riemannian manifold, $\dim(\mathbf{A})$ is naturally
defined.) Then we have

(2) $\dim(\pi_{V}(\mathbf{A}))=s(a),\textup{~{}for~{}a.e.~{}}V\in G(k,n).$

It is not hard to see that

$s(a)\leq\min\\{a,\dim(A(1,k))\\}=\min\\{a,2(k-1)\\}.$

At the first glance, one may guess the optimal number $s(a)$ is
$\min\\{a,2(k-1)\\}$ which has the same form as in (1). However, this is not
true as shown in the following example.

Consider $n=3,k=2$. Let $\mathbf{A}=G(1,3)$ which is the set of lines passing
through the origin. We see that for any $V\in G(2,3)$,
$\pi_{V}(\mathbf{A})=G(1,V)\bigsqcup\\{0\\}$ which is one-dimensional set of
lines plus a point. We get

$\dim(\pi_{V}(\mathbf{A}))=1<2=\min\\{\dim(\mathbf{A}),2(k-1)\\}.$

We see that the problem becomes quite different when we consider the
projection of lines, compared with the projection of points. One reason is
that distinct lines can have overlaps while distinct points do not. Because of
this overlapping phenomenon, we are able to stack lines in $\mathbb{R}^{n}$ so
that their projections to subspaces have some different structure. This makes
the line version of Marstrand projection problem harder than the point
version.

######  Definition 1.

Fix $1<k<n$. For any $0<a<\dim(A(1,n))$, define

(3) $S(a):=\inf_{\mathbf{A}\subset
A(1,n),\dim(\mathbf{A})=a}\operatorname*{ess\,sup}_{V\in
G(k,n)}\dim(\pi_{V}(\mathbf{A})).$

Here $\operatorname*{ess\,sup}$ is with respect to the unique probability
measure on $G(k,n)$ which is invariant under rotation. In our paper, we always
assume $\mathbf{A}$ to be a Borel set to avoid some measurability issue. We
also remark that $S(a)=S_{k,n}(a)$ should also depend on $k,n$, but we just
drop them from the notation for simplicity.

We state our main theorem.

######  Theorem 2.

We have the following exact value of $S(a)$.

(4) $\displaystyle S(a)=a,\ \ \ a\in[0,k-1],$ (5) $\displaystyle S(a)=k-1,\ \
\ a\in[k-1,n-1],$ (6) $\displaystyle S(a)=a-(n-k),\ \ \ a\in[n-1,n+k-2],$ (7)
$\displaystyle S(a)=2(k-1),\ \ \ a\in[n+k-2,2(n-1)].$

######  Remark 3.

We can also write $S(a)$ as

(8) $\displaystyle S(a)=\min\\{a,k-1\\},\ \ \ a\in[0,n-1],$ (9) $\displaystyle
S(a)=\min\\{a-(n-k),2(k-1)\\},\ \ \ a\in[n-1,2(n-1)].$

We will prove Theorem 2 by showing the following two propositions.

######  Proposition 4.

We have the following upper bounds of $S(a)$.

(10) $\begin{split}&S(a)\leq\min\\{a,k-1\\},\ \ \ a\in[0,n-1],\\\
&S(a)\leq\min\\{a-(n-k),2(k-1)\\},\ \ \ a\in[n-1,2(n-1)].\end{split}$

######  Proposition 5.

We have the following lower bounds of $S(a)$.

(11) $\displaystyle S(a)\geq\min\\{a,k-1\\},\ \ \ a\in[0,n-1],$ (12)
$\displaystyle S(a)\geq\min\\{a-(n-k),2(k-1)\\},\ \ \ a\in[n-1,2(n-1)].$

The proof of Proposition 4 is short and provides good examples. We give it
here.

###### Proof of Proposition 4.

When $a\in[0,n-1]$, we choose $\mathbf{A}$ to be a subset of $G(1,n)$ with
dimension $a$. We see that for any $V\in G(k,n)$, $\pi_{V}(\mathbf{A})\subset
G(1,V)\bigsqcup\\{0\\}$. Therefore,

$\dim(\pi_{V}(\mathbf{A}))\leq\min\\{a,k-1\\}$

for any $V\in G(k,n)$, and hence

$S(a)\leq\min\\{a,k-1\\}.$

When $a\in[n-1,2(n-1)]$, we write $a=n-1+\beta$. Choose
$A\subset\mathbb{R}^{n-1}$ (here $\mathbb{R}^{n-1}$ is spanned by the first
$n-1$ coordinates) to be a $\beta$-dimensional set. For each $x\in A$, let
$\mathbf{A}_{x}$ be the set of lines passing through $x$ and transverse to
$\mathbb{R}^{n-1}$. More precisely,

$\mathbf{A}_{x}=\\{\ell\in
A(1,n):x\in\ell,\ell\not\subset\mathbb{R}^{n-1}\\}.$

Choose $\mathbf{A}=\bigsqcup_{x\in A}\mathbf{A}_{x}$. Since
$\\{\mathbf{A}_{x}\\}_{x\in A}$ are disjoint and $\mathbf{A}$ has a product
structure, we have

$\dim(\mathbf{A})=\dim A+\dim(\mathbf{A}_{x})=\beta+n-1=a.$

Note that if $V\in G(k,n)$ and $V$ does not contain the $x_{n}$-axis, then
$\pi_{V}(\mathbf{A}_{x})=\big{(}P_{V}(x)+G(1,V)\big{)}\bigsqcup\\{P_{V}(x)\\}$.
If $V\in G(k,n)$ contains the $x_{n}$-axis, then
$\pi_{V}(\mathbf{A}_{x})=P_{V}(x)+G(1,V)$. As a result,
$\pi_{V}(\mathbf{A})=\bigcup_{x\in A}\pi_{V}(\mathbf{A}_{x})$ consists of at
most $\beta$-dimensional translated copies of $G(1,V)$ plus at most
$\beta$-dimensional set of points. We have

$\dim(\pi_{V}(\mathbf{A}))\leq\beta+k-1=a-(n-k).$

Of course, we also have $\dim(\pi_{V}(\mathbf{A}))\leq\dim(A(1,V)\bigsqcup
V)=2(k-1)$. ∎

For Proposition 5, we will be able to prove a stronger result known as the
exceptional set estimate.

Recall that $P_{V}:\mathbb{R}^{n}\rightarrow V$ is the orthogonal projection
for a given $V\in G(k,n)$. For a set $A\subset\mathbb{R}^{n}$ with $\dim A=a$
and a parameter $s$ satisfying $0<s<\min\\{a,k\\}$, we consider the set

(13) $\\{V\in G(k,n):\dim(P_{V}(A))<s\\},$

which is known as the exceptional set. There are two types of the estimates
for the exceptional set:

(14) $\dim\bigg{(}\\{V\in G(k,n):\dim(P_{V}(A))<s\\}\bigg{)}\leq k(n-k)+s-k.$

(15) $\dim\bigg{(}\\{V\in G(k,n):\dim(P_{V}(A))<s\\}\bigg{)}\leq k(n-k)+s-a;$

We call the first one Kaufman-type estimate, and the second one Falconer-type
estimate. See the reference [7], [2], [10] and [9]. It is not hard to see that
by letting $s\rightarrow\min\\{a,k\\}$, either (15) or (14) will imply (1).

By the same idea, we can deduce Proposition 5 from the following two
exceptional set estimates.

######  Theorem 6.

Fix a number $0<\mu<1/100$. Let $A_{\mu}$ be a ball of radius $\mu$ in
$A_{\text{loc}}(1,n)$ and $G_{\mu}$ be a ball of radius $\mu$ in $G(k,n)$, so
that for any $\ell\in A_{\mu},V\in G_{\mu}$, we have

(16) $\angle(\ell,V^{\perp})>\mu.$

For $\mathbf{A}\subset A_{\mu}$ with $\dim(\mathbf{A})=a$, and
$0<s<\min\\{a,2(k-1)\\}$, define the exceptional set

(17) $E_{s}(\mathbf{A}):=\\{V\in G_{\mu}:\dim(\pi_{V}(\mathbf{A}))<s\\}.$

We have the following estimates:

(18) $\dim(E_{s}(\mathbf{A}))\leq k(n-k)+s-(k-1).$

(19) $\dim(E_{s}(\mathbf{A}))\leq\max\\{0,k(n-k)+s-a+(n-k)\\}.$

######  Remark 7.

For some technical reason, here we use $A_{\mu},G_{\mu}$ (instead of
$A(1,n),G(k,n)$) to make that for any $\ell\in A_{\mu}$ and $V\in G_{\mu}$,
$\pi_{V}(\ell)$ is a line.

The proof of (18) relies on a tube-slab incidence estimate. The proof of (19)
relies on the Fourier analysis in $A(1,n)$, which is the main novelty of this
paper.

There is another notable thing. To prove (1), we just need one of (14) or
(15). However, to prove Proposition 5, we need both (18) and (19). Actually,
(18) implies (11), while (19) implies (12).

Peres and Schlag [10] explored projection problems within a broader context.
For an introduction to this method, we recommend [9, Chapter 18]. Peres and
Schlag introduced the concept of the “transversality condition” for a family
of general projection maps. They also established Marstrand-type estimates and
exceptional set estimates when this condition is met. One might ponder whether
we can employ the approach of Peres and Schlag, involving the definition of
general projection maps and the verification of the transversality condition,
for our specific problem. However, in our case, it does not yield the precise
estimate we require.

In Section 2, we introduce the notation. In Section 3, we show that Theorem 6
implies Proposition 5. In Section 4, we prove the Kaufman-type estimate (18).
In Section 5, we introduce the Fourier transform in $A(1,n)$ and then prove
the Falconer-type estimate (19).

##  2\. Preliminary

We discuss the properties of affine Grassmannians and the metric on it. Most
of the content in this section is from [5, Section 1.2]. See also in [4,
Section 4.1].

###  2.1. Notation and some useful lemmas

We will frequently use the following definitions.

######  Definition 8.

For a number $\delta>0$ and any set $X$ (in a metric space), we use
$|X|_{\delta}$ to denote the maximal number of $\delta$-separated points in
$X$.

######  Definition 9.

Let $\delta,s>0$. We say $A\subset\mathbb{R}^{n}$ is a $(\delta,s,C)$-set if
it is $\delta$-separated and satisfies the following estimate:

(20) $\\#(A\cap B_{r}(x))\leq C(r/\delta)^{s}.$

for any $x\in\mathbb{R}^{n}$ and $1\geq r\geq\delta$. In this paper, the
constant $C$ is not important, so we will just say $A$ is a $(\delta,s)$-set
if

$\\#(A\cap B_{r}(x))\lesssim(r/\delta)^{s}$

for any $x\in\mathbb{R}^{n}$ and $1\geq r\geq\delta$.

######  Remark 10.

We remark that we make “$\delta$-separated” as a part of the definition for a
$(\delta,s)$-set.

######  Lemma 11.

Let $\delta,s>0$ and let $B\subset\mathbb{R}^{n}$ be any set with
$\mathcal{H}^{s}_{\infty}(B)=:\kappa>0$. Then, there exists a $(\delta,s)$-set
$P\subset B$ with $\\#P\gtrsim\kappa\delta^{-s}$.

###### Proof.

See [3, Lemma 3.13]. ∎

######  Lemma 12.

Fix $a>0$. Let $\nu$ be a probability measure satisfying $\nu(B_{r})\lesssim
r^{a}$ for any $B_{r}$ being a ball of radius $r$. If $A$ is a set satisfying
$\nu(A)\geq\kappa$ ($\kappa>0$), then for any $\delta>0$ there exists a subset
$F\subset A$ such that $F$ is a $(\delta,a)$-set and
$\\#F\gtrsim\kappa\delta^{-a}$.

###### Proof.

By the previous lemma, we just need to show
$\mathcal{H}^{a}_{\infty}(A)\gtrsim\kappa$. We just check it by definition.
For any covering $\\{B\\}$ of $A$, we have

$\kappa\leq\sum_{B}\nu(B)\lesssim\sum_{B}r(B)^{a}.$

Ranging over all the covering of $A$ and taking infimum, we get

$\kappa\lesssim\mathcal{H}^{a}_{\infty}(A).$

∎

######  Lemma 13.

Suppose $X\subset[0,1]^{2}$ with $\dim X<s$. Then for any $\varepsilon>0$,
there exist dyadic squares $\mathcal{C}_{2^{-k}}\subset\mathcal{D}_{2^{-k}}$
$(k>0)$ so that

  1. (1)

$X\subset\bigcup_{k>0}\bigcup_{D\in\mathcal{C}_{2^{-k}}}D,$

  2. (2)

$\sum_{k>0}\sum_{D\in\mathcal{C}_{2^{-k}}}r(D)^{s}\leq\varepsilon$,

  3. (3)

$\mathcal{C}_{2^{-k}}$ satisfies the $s$-dimensional condition: For $l<k$ and
any $D\in\mathcal{D}_{2^{-l}}$, we have
$\\#\\{D^{\prime}\in\mathcal{C}_{2^{-k}}:D^{\prime}\subset D\\}\leq
2^{(k-l)s}$.

###### Proof.

See [6, Lemma 2]. ∎

###  2.2. Metric on affine Grassmannian

For every $k$-plane $V\in A(k,n)$, we can uniquely write it as

$V=\textup{dir}(V)+x_{V},$

where $\textup{dir}(V)\in G(k,n)$ and $x_{V}\in V^{\perp}$. $\textup{dir}(V)$
refers to the direction of $V$, as can be seen that
$\textup{dir}(V)=\textup{dir}(V^{\prime})\Leftrightarrow V\parallel
V^{\prime}$.

In this paper, we use $A_{\text{loc}}(k,n)$ to denote the set of $k$-planes
$V$ such that $x_{V}\in B^{n}(0,1/2)$. ($B^{n}(0,1/2)$ is the ball of radius
$1/2$ centered at the origin in $\mathbb{R}^{n}$.)

(21)
$A_{\text{loc}}(k,n)=\\{V:V\textup{~{}is~{}a~{}}k\textup{~{}dimensional~{}plane~{}},x_{V}\in
B^{n}(0,1/2)\\}.$

Later in our proof, instead of considering $A(k,n)$, we only care about those
$V$ lying near the origin. This is through a standard localization argument.

Next, we discuss the metrics on $G(k,n)$ and $A_{\text{loc}}(k,n)$. For
$V_{1},V_{2}\in G(k,n)$, we define

$d(V_{1},V_{2})=\|\pi_{V_{1}}-\pi_{V_{2}}\|.$

Here, $\pi_{V_{1}}:\mathbb{R}^{n}\rightarrow V_{1}$ is the orthogonal
projection. We have another characterization for this metric. Define
$\rho(V_{1},V_{2})$ to be the smallest number $\rho$ such that $B^{n}(0,1)\cap
V_{1}\subset N_{\rho}(V_{2})$. We have the comparability of $d(\cdot,\cdot)$
and $\rho(\cdot,\cdot)$.

######  Lemma 14.

There exists a constant $C>0$ (depending on $k,n$) such that

$\rho(V_{1},V_{2})\leq d(V_{1},V_{2})\leq C\rho(V_{1},V_{2}).$

###### Proof.

Suppose $B^{n}(0,1)\cap V_{1}\subset N_{\rho}(V_{2})$, then for any
$v\in\mathbb{R}^{n}$, we have

$|\pi_{V_{1}}(v)-\pi_{V_{2}}(v)|\lesssim\rho|v|,$

which implies $d(V_{1},V_{2})\lesssim\rho$. On the other hand, if for any
$|v|\leq 1$ we have

$|\pi_{V_{1}}(v)-\pi_{V_{2}}(v)|\leq d|v|,$

then we obtain that $\pi_{V_{1}}(v)\subset N_{d}(V_{2})$. Letting $v$ ranging
over $B^{n}(0,1)\cap V_{1}$, we get $B^{n}(0,1)\cap V_{1}\subset
N_{d}(V_{2})$, which means $\rho(V_{1},V_{2})\leq d$. ∎

We can also define the metric on $A_{\text{loc}}(k,n)$ given by

(22)
$d(V,V^{\prime})=d(\textup{dir}(V),\textup{dir}(V^{\prime}))+|x_{V}-x_{V^{\prime}}|.$

Here, we still use $d$ to denote the metric on $A_{\text{loc}}(k,n)$ and it
will not make any confusion.

Similarly, for $V,V^{\prime}\in A_{\text{loc}}(k,n)$ we can define
$\rho(V,V^{\prime})$ to be the smallest number $\rho$ such that
$B^{n}(0,1)\cap V\subset N_{\rho}(V^{\prime})$. We also have the following
lemma. We left the proof to the interested readers.

######  Lemma 15.

There exists a constant $C>0$ (depending on $k,n$) such that for
$V,V^{\prime}\in A_{\text{loc}}(k,n)$,

$C^{-1}d(V,V^{\prime})\leq\rho(V,V^{\prime})\leq Cd(V,V^{\prime}).$

######  Definition 16.

For $V\in A(k,n)$ and $0<r<1$, we define

$V_{r}:=N_{r}(V)\cap B^{n}(0,1).$

We say that $V_{r}$ is a $k$-dimensional $r$-slab.

Actually, $V_{r}$ is morally a slab of dimensions
$\underbrace{r\times\dots\times
r}_{k\textup{~{}times}}\times\underbrace{1\times\dots\times
1}_{n-k\textup{~{}times}}$. When $k$ is already clear, we simply call $V_{r}$
an $r$-slab. If $W$ is a convex set such that $C^{-1}W\subset V_{r}\subset
CW$, then we also call $W$ an $r$-slab. Here, the constant $C$ will be a fixed
large constant.

######  Definition 17.

For two $r$-slab $V_{r}$ and $V^{\prime}_{r}$. We say they are comparable if
$C^{-1}V_{r}\subset V^{\prime}_{r}\subset CV_{r}$. We say they are essentially
distinct if they are not comparable.

In this paper, we will also consider the balls and $\delta$-neighborhood in
$A_{\text{loc}}(k,n)$. Recall that we use $B_{r}(x)$ to denote the ball in
$\mathbb{R}^{n}$ of radius $r$ centered at $x$. To distinguish the ambient
space, we will use letter $Q$ to denote the balls in $A_{\text{loc}}(k,n)$.
For $V\in A_{\text{loc}}(k,n)$, we use $Q_{r}(V)$ to denote the ball in
$A_{\text{loc}}(k,n)$ of radius $r$ centered at $V$. More precisely,

$Q_{r}(V_{0}):=\\{V\in A_{\text{loc}}(k,n):d(V,V_{0})\leq r\\}.$

For a subset $X\subset A_{\text{loc}}(k,n)$, we use the fancy letter
$\mathcal{N}$ to denote the neighborhood in $A_{\text{loc}}(k,n)$:

$\mathcal{N}_{r}(X):=\\{V\in A_{\text{loc}}(k,n):d(V,X)\leq r\\}.$

Here, $d(V,X)=\inf_{V^{\prime}\in X}d(V,V^{\prime})$.

###  2.3. Hausdorff dimension on metric space

We briefly discuss how to define the Hausdorff dimension for subsets of a
metric space. Let $(M,d)$ be a metric space. For $X\subset M$, we denote the
$s$-dimensional Hausdorff measure of $X$ under the metric $d$ to be
$\mathcal{H}^{s}(X;d)$. We see that if $d^{\prime}$ is another metric on $M$
such that $d(\cdot,\cdot)\sim d^{\prime}(\cdot,\cdot)$, then
$\mathcal{H}^{s}(X;d)\sim\mathcal{H}^{s}(X;d^{\prime})$. It make sense to
define the Hausdorff dimension of $X$ which is independent of the choice of
comparable metrics:

$\dim X:=\sup\\{s:\mathcal{H}^{s}(X;d)>0\\}.$

###  2.4. $\delta$-discretized version: projections of lines onto $k$-planes

We first talk about the projections of points to $k$-planes. Let $x\in
B^{n}(0,1/2)$ and $V$ be a $k$-plane. we see that the orthogonal projection of
$x$ onto $V$ is $P_{V}(x)$, and the fiber of $P_{V}$ at $P_{V}(x)$ is the
$(n-k)$-plane $P_{V}^{-1}(P_{V}(x))$. The $\delta$-discretized version is to
replace $x$ by a $\delta$-ball $B_{\delta}$ centered at $x$, and replace
$P_{V}^{-1}(P_{V}(x))$ by an $(n-k)$-dimensional $\delta$-slab
$T=\bigg{(}P_{V}^{-1}(P_{V}(x))\bigg{)}_{\delta}$. (Here, see Definition 16.)
We see that $T$ is orthogonal to $V$, and $T$ contains $B_{\delta}$.

We want to generalize this $\delta$-discretized notion for projections of
lines. We meet a new issue about transversality: the projection of a line onto
a $k$-plane may be a point but not a line. To handle with this degenerate
case, we need to restrict ourselves to subsets $A_{\mu}\subset
A_{\text{loc}}(1,n)$ and $G_{\mu}\subset G(k,n)$. Here, $0<\mu<1/100$ is a
parameter. $A_{\mu}$ is a $\mu$-ball in $A_{\text{loc}}(1,n)$ and $G_{\mu}$ is
a $\mu$-ball in $G(k,n)$, so that they satisfy the following quantitative
transversality: For any $\ell\in A_{\mu}$ and $V\in G_{\mu}$,

$\angle(\ell,V^{\perp})>\mu.$

(See also (16) in Theorem 6.) Now for any $\ell\in A_{\mu}$ and $V\in
G_{\mu}$, we see that $\ell$ is quantitatively away from $V^{\perp}$, and
hence $P_{V}(\ell)$ is a line in $V$, and there exits an $(n-k+1)$-plane of
form $W=P_{V}(\ell)\oplus W^{\prime}$ where $W^{\prime}$ is an
$(n-k)$-subspace orthogonal to $V$. We say that $W$ is orthogonal to $V$ at
$P_{V}(\ell)$. More generally, we have the following definition.

######  Definition 18.

Let $V\in G(k,n)$, $\ell^{\prime}$ be a line in $V$. If an $(n-k+1)$-plane $W$
is of form $W=\ell^{\prime}\oplus W^{\prime}$ where $W^{\prime}$ is an
$(n-k)$-subspace orthogonal to $V$, then we say that $W$ is orthogonal to $V$
at $\ell^{\prime}$.

######  Remark 19.

Here are two ways to understand this notion in terms of preimage of $P_{V}$,
or preimage of $\pi_{V}$. On the one hand, we see that

$W=P_{V}^{-1}(\ell^{\prime}).$

On the other hand, we can build a subset $\mathbf{W}$ of $A(1,n)$ from $W$ as

$\mathbf{W}:=\\{\ell\in A(1,n):\ell\subset W,\ell\not\perp V\\}.$

Then we also have

$\mathbf{W}=\pi_{V}^{-1}(\ell^{\prime})\cap A(1,n).$

Next, we discuss the geometry of the $\delta$-discretized version. We will
constantly use the following heuristic.

Heuristic. Fix $0<\delta<1$. Given $W\in A_{\text{loc}}(k,n)$ (see (21)), we
have two $\delta$-thickened versions of $W$. One is $W_{\delta}$, which is a
$k$-dimensional $\delta$-slab in $B^{n}(0,1)$. The other is $Q_{\delta}(W)$,
which is a $\delta$-ball in $A_{\text{loc}}(k,n)$. By Lemma 15 and ignoring
some constant, we can morally think of $W_{\delta}$ and $Q_{\delta}(W)$ as
follows:

(23) $\begin{split}Q_{\delta}(W)&\approx\\{W^{\prime}\in
A_{\text{loc}}(k,n):W^{\prime}\cap B^{n}(0,1)\subset W_{\delta}\\},\\\
W_{\delta}&\approx\bigg{(}\bigcup_{W^{\prime}\in
Q_{\delta}(W)}W^{\prime}\bigg{)}\cap B^{n}(0,1).\end{split}$

It is good to think of them as the same thing. The reader can consider the two
$\delta$-thickened versions for a point $x$: both of $Q_{\delta}(\\{x\\})$ and
$\\{x\\}_{\delta}$ are $B^{n}(x,\delta)$ (the ball of radius $\delta$ centered
at $x$), since we can identify $A_{\text{loc}}(0,n)$ as a subset of
$\mathbb{R}^{n}$.

Let us talk about the projections. As before, let $\ell\in A_{\mu}$ and $V\in
G_{\mu}$. We have two $\delta$-thickened versions of $\ell$: $\ell_{\delta}$
which is a subset of $B^{n}(0,1)$, and $Q_{\delta}(\ell)$ which is a subset of
$A_{\text{loc}}(1,n)$. By the quantitative transversality condition between
$\ell$ and $V$, we see that $P_{V}(\ell_{\delta})$ has dimensions
$\sim\delta\times\dots\times\delta\times 1$, where the implicit constant
depends on $\mu$. When $\mu$ is fixed, we may just ignore this implicit
constant, so let us assume $P_{V}(\ell_{\delta})$ is a $\delta$-tube.

We have the $\delta$-disretized version for Definition 18. Let
$W=P_{V}(\ell)\oplus W^{\prime}$ which is an $(n-k+1)$-plane orthogonal to $V$
at $P_{V}(\ell)$. We can morally think of $W_{\delta}$ as
$P_{V}(\ell_{\delta})\times W^{\prime}$. We can also morally think of
$W_{\delta}$ as $P_{V}^{-1}(P_{V}(\ell_{\delta}))\cap B^{n}(0,1)$. There are
some features of $W_{\delta}$:

  1. (1)

It is an $(n-k+1)$-dimensional $\delta$-slab;

  2. (2)

It contains $\ell_{\delta}$;

  3. (3)

Its intersection with $V$ is a $\delta$-tube $P_{V}(\ell_{\delta})$, and the
other $(n-k)$ directions are orthogonal to $V$.

We would like to say $W_{\delta}$ is orthogonal to $V$ at
$P_{V}(\ell_{\delta})$. See Figure 1.

Figure 1.

##  3\. Theorem 6 implies Proposition 5

We will use the Kaufman-type estimate (18) to deduce (11), and use Falconer-
type estimate (19) to deduce (12).

We first prove (11). Since $S(a)$ is monotonically increasing in $a$, we can
assume $a\in[0,k-1]$ and to prove

$S(a)\geq a.$

Suppose by contradiction that $S(a)\leq a-2\varepsilon$ for some
$\varepsilon>0$. From the definition of $S(a)$ in (3), we can find a set
$\mathbf{A}\subset A(1,n)$ with $\dim(\mathbf{A})=a$ so that the set

(24) $E=\\{V\in G(k,n):\dim(\pi_{V}(\mathbf{A}))<a-\varepsilon\\}$

has positive measure. Since $A(1,n)$ can be covered by countably many
translated copies of $A_{\text{loc}}(1,n)$, so we may assume
$\mathbf{A}\subset A_{\text{loc}}(1,n)$ with
$\dim(\mathbf{A})>a-\varepsilon/100$ and (24) still has positive measure.

Since $\dim(\mathbf{A})>a-\varepsilon/100$, there exists $\ell\in\mathbf{A}$
such that

$\dim(Q_{r}(\ell)\cap\mathbf{A})\geq a-\varepsilon/2,$

for any $r>0$ and $Q_{r}(\ell)$ being a ball of radius $r$ centered at $\ell$
in the metric space $A_{\text{loc}}(1,n)$. Usually, $\ell$ is referred to as
an $(a-\varepsilon/2)$-density point in $\mathbf{A}$. Without loss of
generality, we assume $\ell$ is parallel to the $x_{n}$-axis. We also use
$\ell_{0}$ to denote the translation of $\ell$ to the origin, which by our
assumption, is the $x_{n}$-axis.

Since $\\{V\in G(k,n):\ell_{0}\subset V^{\perp}\\}$ has zero measure as a
subset of $G(k,n)$, we see that

$E\cap\\{V\in G(k,n):\ell_{0}\not\subset V^{\perp}\\}$

has positive measure. For any $V\in\\{V\in G(k,n):\ell_{0}\subset
V^{\perp}\\}$, there exists a number $\mu=\mu_{V}>0$ such that the following
holds. Let $Q_{\mu}(\ell)\subset A_{\text{loc}}(1,n)$ be a ball of radius
$\mu$ centered at $\ell$, and $Q_{\mu}(V)\subset G(k,n)$ be a ball of radius
$\mu$ centered at $V$. Then

$\angle(\ell_{1},V_{1}^{\perp})>\mu$

for any $\ell_{1}\in Q_{\mu}(\ell)$ and $V_{1}\in Q_{\mu}(V)$. This guarantees
the transversality condition (16). We will let $s=a-\varepsilon$. We can check

$s<\min\\{\dim(Q_{r}(\ell)\cap\mathbf{A})),2(k-1)\\}.$

Then, we can apply Theorem 6 to the set $\mathbf{A}\cap Q_{\mu}(\ell)$. By
(18), we obtain that

(25) $\begin{split}\dim(\\{V\in Q_{\mu}(V):\dim(\pi_{V}(\mathbf{A}\cap
Q_{\mu}(\ell)))<a-\varepsilon\\})&\leq k(n-k)+a-\varepsilon-(k-1)\\\ &\leq
k(n-k)-\varepsilon.\end{split}$

The last inequality is because $a\leq k-1$. This implies

(26) $\dim(\\{V\in Q_{\mu}(V):\dim(\pi_{V}(\mathbf{A}))<a-\varepsilon\\})\leq
k(n-k)-\varepsilon.$

Since we can write $E\cap\\{V\in G(k,n):\ell_{0}\not\subset V^{\perp}\\}$ as a
countable union of the set of form

$\\{V\in Q_{\mu}(V):\dim(\pi_{V}(\mathbf{A}))<a-\varepsilon\\}.$

Therefore, $\dim(E\cap\\{V\in G(k,n):\ell_{0}\not\subset V^{\perp}\\})\leq
k(n-k)-\varepsilon$, which contradicts that $E\cap\\{V\in
G(k,n):\ell_{0}\not\subset V^{\perp}\\}$ has positive measure.

Next. we prove (12). We can assume $a\in[n-1,n+k-2]$ and to prove

$S(a)\geq a-(n-k).$

Arguing in the same way, we can find a set $\mathbf{A}\subset
A_{\text{loc}}(1,n)$ with $\dim(\mathbf{A})>a-\varepsilon/100$ so that

$E=\\{V\in G(k,n):\dim(\pi_{V}(\mathbf{A}))<a-(n-k)-\varepsilon\\}$

has positive measure. By finding an $(a-\varepsilon/2)$-density point $\ell$
in $\mathbf{A}$ and similarly defining $\ell_{0}$, we can assume

$E\cap\\{V\in G(k,n):\ell_{0}\not\subset V^{\perp}\\}$

has positive measure. We will let $s=a-(n-k)-\varepsilon$. We can check

$s<\min\\{a-\varepsilon/2,2(k-1)\\}.$

Then, we can apply Theorem 6 and obtain a similar estimate as (26):

(27) $\begin{split}&\ \ \ \dim(\\{V\in
Q_{\mu}(V):\dim(\pi_{V}(\mathbf{A}))\leq a+(n-k)-\varepsilon\\})\\\ &\leq
k(n-k)+(a-(n-k)-\varepsilon)-(a-\varepsilon/2)+(n-k)\\\ &\leq
k(n-k)-\varepsilon/2.\end{split}$

Therefore, $\dim(E\cap\\{V\in G(k,n):\ell_{0}\not\subset V^{\perp}\\})\leq
k(n-k)-\varepsilon/2$, which contradicts that $E\cap\\{V\in
G(k,n):\ell_{0}\not\subset V^{\perp}\\}$ has positive measure.

######  Remark 20.

We actually showed that the number

(28) $\widetilde{S}(a):=\inf_{\mathbf{A}\subset
A(1,n),\dim(\mathbf{A})=a}\operatorname*{ess\,sup}_{V\in
G(k,n)}\dim\Big{(}\pi_{V}(\mathbf{A})\cap A(1,V)\Big{)},$

which looks smaller than $S(a)$ defined in (3), actually has the same lower
bound as $S(a)$ in Proposition 5.

##  4\. Kaufman-type exceptional set estimate

We state a discretized version of the estimate (18).

######  Theorem 21.

Fix a number $0<\mu<1/100$, and let $A_{\mu},G_{\mu}$ be as in Theorem 6. Fix
$t>k(n-k)-(k-1)$, $0<s<a$ and $0<u<t-k(n-k)+(k-1)$. For sufficiently small
$\varepsilon>0$ (depending on $\mu,n,k,a,s,t,u$), the following holds. Let
$0<\delta<1/100$. Let $\mathbf{H}\subset A_{\mu}$ be a $(\delta,u)$-set with
$\\#\mathbf{H}\gtrsim(\log\delta^{-1})^{-2}\delta^{-u}$. Let
$\mathcal{V}\subset G_{\mu}$ be a $(\delta,t)$-set with
$\\#\mathcal{V}\gtrsim(\log\delta^{-1})^{-2}\delta^{-t}$. Suppose for each
$V\in G(k,n)$, we have a collection of slabs $\mathbb{T}_{V}$, where each
$T\in\mathbb{T}_{V}$ has dimensions
$\underbrace{\delta\times\dots\times\delta}_{k-1\
\textup{times}}\times\underbrace{1\times\dots\times 1}_{n-k+1\
\textup{times}}$ and is orthogonal to $V$ at some $\delta$-tube in $V$.

The $s$-dimensional Forstman condition holds for $\mathbb{T}_{V}$: for any
$\delta\leq r\leq 1$ and any $(n-k+1)$-dimensional $r$-slab $W_{r}$ that is
orthogonal to $V$ at some $r$-tube in $V$, we have

(29) $\\#\\{T\in\mathbb{T}_{V}:T\subset W_{r}\\}\lesssim(r/\delta)^{s}.$

We also assume that for each $\ell\in\mathbf{H}$,

(30) $\\#\\{V\in\mathcal{V}:\ell_{\delta}\subset
T,\textup{~{}for~{}some~{}}T\in\mathbb{T}_{V}\\}\gtrsim(\log\delta^{-1})^{-2}\\#\mathcal{V}.$

Then, we have

(31)
$\delta^{-u-t}\lesssim_{\varepsilon}\delta^{-\varepsilon/2}\sum_{V\in\mathcal{V}}\\#\mathbb{T}_{V}\lesssim\delta^{-\varepsilon-
t-s}.$

###  4.1. Proof of the Kaufman-type estimate

We give the proof of (18) using Theorem 21. We choose
$\alpha<\dim(\mathbf{A}),t<\dim(E_{s}(\mathbf{A}))$. By Frostman’s lemma,
there exist probability measures $\nu_{\mathbf{A}}$ supported on $\mathbf{A}$
and $\nu_{E}$ supported on $E_{s}(\mathbf{A})$ satisfying the Frostman’s
condition:

(32) $\displaystyle\nu_{\mathbf{A}}(Q_{r})\lesssim
r^{\alpha}\textup{~{}for~{}any~{}}Q_{r}\textup{~{}being~{}a~{}ball~{}of~{}radius~{}}r\textup{~{}in~{}}A_{\text{loc}}(1,n).$
(33) $\displaystyle\nu_{E}(Q_{r})\lesssim
r^{t}\textup{~{}for~{}any~{}}Q_{r}\textup{~{}being~{}a~{}ball~{}of~{}radius~{}}r\textup{~{}in~{}}G(k,n).$

We only need to prove

(34) $t\leq k(n-k)+s-(k-1),$

since then we can let $t\rightarrow\dim(E_{s}(\mathbf{A}))$.

Fix a $V\in E_{s}(\mathbf{A})$. By definition, we have
$\dim(\pi_{V}(\mathbf{A}))<s$. We also fix a small number $\epsilon_{\circ}$
which we will later send to $0$. We view $\pi_{V}(\mathbf{A})$ as a subset of
$A_{\text{loc}}(1,V)=A_{\text{loc}}(1,k)$. By Lemma 13, we can cover
$\pi_{V}(\mathbf{A})$ by balls $\mathbb{D}_{V}=\\{D\\}$ in
$A_{\text{loc}}(1,V)$, each of which has radius $2^{-j}$ for some integer
$j>|\log_{2}\epsilon_{\circ}|$. We define
$\mathbb{D}_{V,j}:=\\{D\in\mathbb{D}_{V}:r(D)=2^{-j}\\}$. Lemma 13 yields the
following properties:

(35) $\sum_{D\in\mathbb{D}_{V}}r(D)^{s}<1,$

and for each $j$ and $r$-ball $Q_{r}\subset A_{\text{loc}}(1,V)$, we have

(36) $\\#\\{D\in\mathbb{D}_{V,j}:D\subset
Q_{r}\\}\lesssim\left(\frac{r}{2^{-j}}\right)^{s}.$

On the one hand, $D$ is a $2^{-j}$-ball in $A_{\text{loc}}(1,V)$. On the other
hand we can view $D$ as a $2^{-j}$-tube in $V$. We use $t_{D}$ to denote this
$2^{-j}$-tube. By the heuristic (23), we can view $t_{D}$ as

$\bigcup_{\ell\in D}\ell\cap\\{x\in V:|x|\leq 1\\}.$

For each $V\in E_{s}(A)$, we can find such a
$\mathbb{D}_{V}=\bigcup_{j}\mathbb{D}_{V,j}$. We also define the slabs
$\mathbb{T}_{V,j}:=\\{\pi^{-1}_{V}(t_{D})\cap
B^{n}(0,1):D\in\mathbb{D}_{V,j}\\}$,
$\mathbb{T}_{V}=\bigcup_{j}\mathbb{T}_{V,j}$. Each plank in $\mathbb{T}_{V,j}$
has dimensions

$\underbrace{2^{-j}\times 2^{-j}\times\dots\times
2^{-j}}_{k-1\text{~{}times}}\times\underbrace{1\times 1\times\dots\times
1}_{n-k+1\text{~{}times}}$

such that it is orthogonal to $V$ at some $t_{D}$ for $D\in\mathbb{D}_{V,j}$.
For each such slab $T\in\mathbb{T}_{V,j}$, we use its bold-font $\mathbf{T}$
to denote the set of lines whose unit truncations are in $T$. More precisely,

(37) $\mathbf{T}:=\\{\ell:\ell\cap B^{n}(0,1)\subset T\\}.$

One easily sees that
$\mathbf{A}\subset\bigcup_{T\in\mathbb{T}_{V}}\mathbf{S}$. By pigeonholing,
there exists $j(V)$ such that

(38)
$\nu_{\mathbf{A}}(\mathbf{A}\cap(\cup_{T\in\mathbb{T}_{V,j(V)}}\mathbf{T}))\geq\frac{1}{10j(V)^{2}}\nu_{\mathbf{A}}(\mathbf{A})=\frac{1}{10j(V)^{2}}.$

For each $j>|\log_{2}\epsilon_{\circ}|$, define $E_{s,j}(A):=\\{V\in
E_{s}(A):j(V)=j\\}$. Then we obtain a partition of $E_{s}(\mathbf{A})$:

$E_{s}(\mathbf{A})=\bigsqcup_{j}E_{s,j}(\mathbf{A}).$

By pigeonholing again, there exists $j$ such that

(39)
$\nu_{E}(E_{s,j}(\mathbf{A}))\geq\frac{1}{10j^{2}}\nu_{E}(E_{s}(\mathbf{A}))=\frac{1}{10j^{2}}.$

In the rest of the proof, we fix this $j$. We also set $\delta=2^{-j}$. By
Lemma 12, there exists a $(\delta,t)$-set $\mathcal{V}\subset
E_{s,j}(\mathbf{A})$ with cardinality
$\\#\mathcal{V}\gtrsim(\log\delta^{-1})^{-2}\delta^{-t}$.

Next, we consider the set
$M:=\\{(\ell,V)\in\mathbf{A}\times\mathcal{V}:\ell\in\cup_{T\in\mathbb{T}_{V,j}}\mathbf{T}\\}$.
We also use $\\#$ to denote the counting measure on $\mathcal{V}$. Define the
sections of $M$:

$M_{\ell}=\\{V:(\ell,V)\in M\\},\ \ \ M_{V}:=\\{\ell:(\ell,V)\in M\\}.$

By (38) and Fubini, we have

(40) $(\nu_{\mathbf{A}}\times\\#)(M)\geq\frac{1}{10j^{2}}\mu(\mathcal{V}).$

This implies

(41) $(\nu_{\mathbf{A}}\times\\#)\bigg{(}\Big{\\{}(\ell,V)\in
M:\mu(M_{\ell})\geq\frac{1}{20j^{2}}\mu(\mathcal{V})\Big{\\}}\bigg{)}\geq\frac{1}{20j^{2}}\mu(\mathcal{V}).$

By (41), we have

(42)
$\nu_{\mathbf{A}}\bigg{(}\Big{\\{}\ell\in\mathbf{A}:\mu(M_{\ell})\geq\frac{1}{20j^{2}}\mu(\mathcal{V})\Big{\\}}\bigg{)}\geq\frac{1}{20j^{2}}.$

We are ready to apply Theorem 21. Recall $\delta=2^{-j}$ and
$\\#\mathcal{V}\gtrsim(\log\delta^{-1})^{-2}\delta^{-t}$. We may assume
$t>k(n-k)-(k-1)$, otherwise we are done. Set

(43) $u=\min\\{t-k(n-k)+(k-1),a\\}-\varepsilon.$

By (42) and Lemma 12, we can find a $(\delta,u)$-subset of
$\\{\ell\in\mathbf{A}:\\#M_{\ell}\geq\frac{1}{20j^{2}}\\#\mathcal{V}\\}$ with
cardinality $\gtrsim(\log\delta^{-1})^{-2}\delta^{-u}$. We denote this set by
$\mathbf{H}$. For each $\ell\in H$, we see that there are
$\gtrsim(\log\delta^{-1})^{-2}\\#\mathcal{V}$ many slabs from
$\cup_{V\in\mathcal{V}}\mathbb{T}_{V,j}$ that contain $\ell_{\delta}$. We can
now apply Theorem 21 to obtain

$\delta^{-u-t}\lesssim_{\varepsilon}\delta^{-\varepsilon-t-s}.$

By letting $\epsilon_{\circ}\rightarrow 0$ (and hence $\delta\rightarrow 0$),
we obtain

$u+t\leq t+s+\varepsilon.$

Plugging in the definition of $u$ and letting $\varepsilon\rightarrow 0$, we
obtain

$\min\\{t-k(n-k)+(k-1),a\\}\leq s.$

Since $a>s$, we obtain

(44) $t\leq k(n-k)+s-(k-1).$

###  4.2. Discretized Kaufman-type estimate

###### Proof of Theorem 21.

Define $\mathbb{T}=\bigcup_{V\in\mathcal{V}}\mathbb{T}_{V}$, where
$\mathbb{T}_{V}$ is given by Theorem 21. If two $\delta$-slab are comparable,
then we just identify them. So we assume the $\delta$-slabs in $\mathbb{T}$
are essentially distinct.

We define the following incidence pair:

(45)
$\mathcal{I}=\mathcal{I}(\mathbb{T},\mathcal{V}):=\\{(T,V)\in\mathbb{T}\times\mathcal{V}:T\in\mathbb{T}_{V}\\}.$

We will prove the theorem by comparing the upper and lower bound of
$\mathcal{I}$. We easily see the upper bound

(46)
$\\#\mathcal{I}\lesssim\sum_{V\in\mathcal{V}}\\#\mathbb{T}_{V}\lesssim\\#\mathcal{V}\\#\mathbb{T}_{V}\lesssim\delta^{-t-s}.$

Here, we used the bound $\\#\mathcal{V}\lesssim\delta^{-t}$ since
$\mathcal{V}$ is a $(\delta,t)$-set, and we used
$\\#\mathbb{T}_{V}\lesssim\delta^{-s}$ by plugging in $r=1$ in (29).

For the lower bound of $\\#\mathcal{I}$, we will use the following inequality:

$\\#(\bigcup_{i}A_{i})\geq\sum_{i}\\#(A_{i}\setminus\bigcup_{j\neq i}A_{j}).$

By [1, Lemma 13], we choose a $\delta|\log\delta|^{O(1)}$-separated subset
$\mathbf{H}^{\prime}\subset\mathbf{H}$ such that $\mathbf{H}^{\prime}$ is a
$(\delta|\log\delta|^{O(1)},u)$-set and

(47) $\\#\mathbf{H}^{\prime}\gtrsim|\log\delta|^{-O(1)}\delta^{-u}.$

Here, $O(1)$ is a large constant to be determined later.

For each $\ell\in\mathbf{H}^{\prime}$, we define the following subset of
$\mathcal{I}$.

(48) $\mathcal{I}_{\ell}:=\\{(T,V)\in\mathcal{I}:\ell_{\delta}\subset T\\}.$

Then,
$\mathcal{I}\supset\bigcup_{\ell\in\mathbf{H}^{\prime}}\mathcal{I}_{\ell}$.

We have

(49)
$\displaystyle\\#\mathcal{I}\geq\\#\left(\bigcup_{\ell\in\mathbf{H}^{\prime}}\mathcal{I}_{\ell}\right)$
$\displaystyle\geq\\#\left(\bigcup_{\ell\in\mathbf{H}^{\prime}}\left(\mathcal{I}_{\ell}\setminus\bigcup_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\mathcal{I}_{\ell_{1}}\right)\right)$
(50)
$\displaystyle=\sum_{\ell\in\mathbf{H}^{\prime}}\\#\left(\mathcal{I}_{\ell}\setminus\bigcup_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\mathcal{I}_{\ell_{1}}\right)$
(51)
$\displaystyle=\sum_{\ell\in\mathbf{H}^{\prime}}\bigg{(}\\#\mathcal{I}_{\ell}-\sum_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\\#\left(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}}\right)\bigg{)}.$

We will show that

$\\#\mathcal{I}_{\ell}-\sum_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\\#\left(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}}\right)\geq\frac{1}{2}\\#\mathcal{I}_{\ell}.$

This will imply

(52)
$\\#\mathcal{I}\geq\frac{1}{2}\sum_{\ell\in\mathbf{H}^{\prime}}\\#\mathcal{I}_{\ell}\gtrsim(\log\delta^{-1})^{-O(1)}\delta^{-u-t}.$

We make one observation for $\mathcal{I}_{\ell}$. Note that for any
$V\in\mathcal{V}$, there are $\lesssim 1$ $T_{i}$’s from $\mathbb{T}$ such
that $(T_{i},V)\in\mathcal{I}_{\ell}$. The reason is that if
$T_{i}\in\mathcal{I}_{\ell}$, then $T_{i}$ is orthogonal to $V$ at
$P_{V}(\ell_{\delta})$. There can be at most $\lesssim 1$ such $T_{i}$’s. By
losing a constant factor in the estimate, we may assume for any
$V\in\mathcal{V}$ there is one or no $T$ such that
$(T,V)\in\mathcal{I}_{\ell}$. Therefore, we can identify $\mathcal{I}_{\ell}$
with the set on the left hand side of (30). Also, (30) implies for any
$\ell\in\mathbf{H}^{\prime}$,

(53)
$\\#\mathcal{I}_{\ell}\gtrsim(\log\delta^{-1})^{-2}\\#\mathcal{V}\gtrsim(\log\delta^{-1})^{-4}\delta^{-t}.$

For fixed $\ell$, and $\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}$, we
want to find an upper bound for
$\\#(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}})$.

By the condition that $\mathbf{H}^{\prime}$ is contained in a $\mu$-ball in
$A_{\text{loc}}(1,n)$, we may assume all the $\ell$ in $\mathbf{H}^{\prime}$
form an angle $\leq 10^{-1}$ with the $x_{n}$-axis. We will introduce a new
metric for these lines. It will not be hard to see that this metric is
comparable to the metric on $A_{\text{loc}}(1,n)$ introduced in (22).

Let
$\Pi_{0}=\mathbb{R}^{n-1}\times\\{0\\},\Pi_{1}=\mathbb{R}^{n-1}\times\\{1\\}$.
Then each $\ell\in\mathbf{H}^{\prime}$ will intersects $\Pi_{0}$, $\Pi_{1}$ at
two points denoted by $P_{0}(\ell),P_{1}(\ell)$. We will use
$(P_{0}(\ell),P_{1}(\ell))$ for the local coordinates of $\ell$. And we define
the metric to be

(54)
$d(\ell,\ell_{1}):=|P_{0}(\ell)-P_{0}(\ell_{1})|+|P_{1}(\ell)-P_{1}(\ell_{1})|.$

This metric is equivalent to the metric defined in (22). We leave out the
proof.

Next, we will estimate $\\#(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}})$. By
the definition of $d(\ell,\ell_{1})$, we may assume
$|P_{0}(\ell)-P_{0}(\ell_{1})|\geq\frac{1}{2}d(\ell,\ell_{1})$. We denote the
number

$d:=|P_{0}(\ell)-P_{0}(\ell_{1})|.$

We remind the reader $\delta/2\leq d\leq 1$.

We introduce the two dimensional plank $\mathcal{P}$ which has size $\sim
1\times d$. The side of $\mathcal{P}$ of length $1$ is $\ell\cap B^{n}(0,1)$.
The side of $\mathcal{P}$ of length $d$ is
$\overrightarrow{P_{0}(\ell)P_{0}(\ell_{1})}$.

If $(T,V)\in\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}}$, then
$\mathcal{P}\subset T$ and $P_{V}(T)$ is contained in a $\delta$-tube.
Therefore, $P_{V}(\mathcal{P})$ is contained in a $\delta$-tube. We formally
write

$\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}}\supset\\{V\in\mathcal{V}:P_{V}(\mathcal{P})\textup{~{}is~{}contained~{}in~{}a~{}}\delta\textup{-tube}\\}.$

We make the following observation.

######  Lemma 22.

Let $\mathcal{M}$ be a set of $k$-dimensional subspaces $V$ so that the
projection of $\mathcal{P}$ to $V$ is a line or point. In other words,

(55) $\mathcal{M}:=\\{V\in G(k,n):\dim(P_{V}(\mathcal{P}))\leq 1\\}.$

Then we have

(56) $\\{V\in G(k,n):P_{V}(\mathcal{P})\textup{~{}is contained in
a~{}}\delta\textup{-tube}\\}\subset\mathcal{N}_{C\delta d^{-1}}(\mathcal{M}).$

Here, $C$ is some big constant.

###### Proof.

We sketch the idea of the proof. Let $V$ be in the LHS of (56). Then
$P_{V}(\mathcal{P})$ is contained in a $1\times\delta$-tube. Define
$\mathcal{Q}$ to be a $1\times 1$-square, which is obtained by prolonging the
length-$d$ side of $\mathcal{P}$ $d^{-1}$ times. We see that
$P_{V}(\mathcal{Q})$ is contained in a $1\times d^{-1}\delta$-tube. The next
step is a rotation argument. We claim that by rotating $V$ within angle
$\lesssim d^{-1}\delta$, we obtain another $k$-subspaces $W$ such that
$P_{W}(\mathcal{Q})$ is a line. We leave out the proof of this claim.

Therefore, we find a $W\in\mathcal{M}$ such that $d(V,W)\lesssim
d^{-1}\delta$, which finishes the proof of lemma. ∎

The next lemma is about the dimension of $\mathcal{M}$.

######  Lemma 23.

Let $\mathcal{M}$ be given by (55). Then

(57) $\dim(\mathcal{M})=1+k(n-k-1).$

###### Proof.

We sketch the proof. We may assume $\mathcal{P}$ span the plane
$\mathbb{R}^{2}$. If $\dim(P_{V}(\mathbb{R}^{2}))\leq 1$, then
$V^{\perp}\cap\mathbb{R}^{2}$ contains a line or the whole $\mathbb{R}^{2}$.
We just need to consider the first case. We think of $V^{\perp}$ as spanned by
$n-k$ orthonormal vectors $v_{1},\dots,v_{n-1}$. From the condition that
$V^{\perp}\cap\mathbb{R}^{2}$ contains a line, we have one dimensional choice
of vector $V_{1}$ that lies in $\mathbb{R}^{2}$, and then we choose
$v_{2},\dots,v_{n-k}$ from $v_{1}^{\perp}$ which has
$\dim(G(n-k-1,n-1))=k(n-k-1)$ dimensional choice. Therefore, we see that such
$V^{\perp}$ and hence $V$ has $1+k(n-k-1)$ dimensional choice. ∎

We have the estimate

(58)
$\\#(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}})\leq\\#(\mathcal{V}\cap\mathcal{N}_{C\delta
d^{-1}}(\mathcal{M})).$

And we remind the reader that $\mathcal{V}$ is a $(\delta,t)$-set.

By Lemma 23, we can cover $\mathcal{N}_{C\delta d^{-1}}(\mathcal{M})$ by
$\sim(\delta^{-1}d)^{1+k(n-k-1)}$ many $\delta d^{-1}$-balls in $G(k,n)$. By
the $(\delta,t)$ property of $\mathcal{V}$, we have

$\\#(\mathcal{V}\cap\mathcal{N}_{C\delta
d^{-1}}(\mathcal{M}))\lesssim(\delta^{-1}d)^{1+k(n-k-1)}d^{-t}.$

So, we have

$\displaystyle\sum_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\\#(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}})$
$\displaystyle\lesssim\sum_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\left(\delta^{-1}d(\ell,\ell_{1})\right)^{1+k(n-k-1)}d(\ell,\ell_{1})^{-t}$
$\displaystyle=\sum_{\delta|\log\delta|^{O(1)}\leq d\leq 1}\ \
\sum_{\ell_{1}\in\mathbf{H}^{\prime},d(\ell,\ell_{1})\sim
d}\left(\delta^{-1}d\right)^{1+k(n-k-1)}d^{-t}.$

Here the summation over $d$ is over dyadic numbers. Since
$\\#(\mathbf{H}^{\prime}\cap Q_{d}(\ell))\lesssim(\frac{d}{\delta})^{u}$, the
expression above is bounded by

$\displaystyle\lesssim\sum_{\delta|\log\delta|^{O(1)}\leq d\leq
1}\left(\frac{d}{\delta}\right)^{u}\left(\frac{d}{\delta}\right)^{1+k(n-k-1)}d^{-t}$
$\displaystyle=\delta^{-t}\sum_{\delta|\log\delta|^{O(1)}\leq d\leq
1}\left(\frac{d}{\delta}\right)^{u+1+k(n-k-1)-t}$
$\displaystyle\lesssim\delta^{-t}|\log\delta|^{O(1)(u+1+k(n-k-1)-t)}.$

From the condition in Theorem 21, $u+1+k(n-k-1)-t\leq-\varepsilon<0$. by
choosing the constant $O(1)$ big enough, we have

$\sum_{\ell_{1}\in\mathbf{H}^{\prime}\setminus\\{\ell\\}}\\#(\mathcal{I}_{\ell}\cap\mathcal{I}_{\ell_{1}})\leq
C^{-1}|\log\delta|^{-4}\delta^{-t}\leq\frac{1}{2}\\#\mathcal{I}_{\ell}.$

As a result, we have

$\\#\mathcal{I}\geq|\log\delta|^{-O(1)}\delta^{-u-t}.$

Compared with the upper bound of $\\#\mathcal{I}$ in (46), we finish the
proof. ∎

##  5\. Falconer-type exceptional set estimate

We will use the high-low method to prove Falconer-type exceptional set
estimate. The key ingredient in the high-low method is the Fourier analysis.
Since we are working with the set $\mathbf{A}$ as a subset of $A(1,n)$, we
need the Fourier transform in $A(1,n)$. It is quite hard to define a global
Fourier transform on $A(1,n)$, but since our set $\mathbf{A}$ has been
localized in $A_{\mu}$ (see Theorem 6), we just need to consider the Fourier
transform in the set $\widetilde{A}(1,n)$ consisting of lines in $A(1,n)$ that
is transverse to $\mathbb{R}^{n-1}$. In other words,

(59) $\widetilde{A}(1,n):=\\{\ell\in
A(1,n):\ell\textup{~{}is~{}not~{}parallel~{}to~{}}\mathbb{R}^{n-1}\\}.$

In the next subsection, we introduce the Fourier transform in
$\widetilde{A}(1,n)$ and discuss some properties.

###  5.1. Fourier transform on $A(1,n)$

First, we introduce the coordinate for $\widetilde{A}(1,n)$. By definition,
every $\ell\in\widetilde{A}(1,n)$ intersect $\mathbb{R}^{n-1}\times\\{0\\}$
and $\mathbb{R}^{n-1}\times\\{1\\}$ at two points for which we denote by
$X(\ell),Y(\ell)$. We see that we parametrize $\widetilde{A}(1,n)$ using the
coordinates $(X,Y)$.

(60)
$(X,Y):\widetilde{A}(1,n)\xrightarrow{\cong}\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}\
\ \ell\mapsto(X(\ell),Y(\ell)).$

Here, we can just view $\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$ as
$\mathbb{R}^{2(n-1)}$. We see that we can pull back the Fourier transform in
$\mathbb{R}^{2(n-1)}$ to that in $\widetilde{A}(1,n)$. For convenience, we
denote the inverse of (60) to be

(61)
$\ell:\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}\xrightarrow{\cong}\widetilde{A}(1,n)\
\ (X,Y)\mapsto\ell(X,Y).$

Here is the precise definition for Fourier transform on $\widetilde{A}(1,n)$.
Suppose $F=F(\ell)$ is a function on $\widetilde{A}(1,n)$, we define the
Fourier transform of $F$, denoted by $\widehat{F}(\ell)$, to be the function
on $\widetilde{A}(1,n)$ given by

(62)
$\widehat{F}(\ell):=\big{(}\mathcal{F}(F\circ\ell)\big{)}(X(\ell),Y(\ell)).$

We explain the expression. Given $F$ as a function on $\widetilde{A}(1,n)$, we
use $\ell$ in (61) to pull back $F$ to a function $F\circ\ell$ which is
defined on $\mathbb{R}^{2(n-1)}$. We apply $\mathcal{F}$, which is the
standard Fourier transform on $\mathbb{R}^{2(n-1)}$, to $F\circ\ell$, and
obtain the function $\mathcal{F}(F\circ\ell)$ on $\mathbb{R}^{2(n-1)}$. At
last, we use $(X,Y)$ in (60) to pull back $\mathcal{F}(F\circ\ell)$ to the
function defined on $\widetilde{A}(1,n)$.

Next, we will introduce the dual rectangle. This tool is always used as a
black box. It says that if a function is a smooth bump function adapted to a
rectangle, then its Fourier transform is morally a smooth function adapted to
the dual rectangle multiplied by some normalizing constant. We first review
this property for Fourier transform in $\mathbb{R}^{n}$ and then talk about it
in $\widetilde{A}(1,n)$.

######  Definition 24.

Let $R\subset\mathbb{R}^{n}$ be a rectangle of dimensions $a_{1}\times
a_{1}\times\dots\times a_{n}$. We define the dual rectangle of $R$ to be
another rectangle $R^{*}$ centered at the origin with dimensions
$a_{1}^{-1}\times a_{2}^{-1}\times\dots\times a_{n}^{-1}$, so that the edge of
$R^{*}$ of length $a_{i}^{-1}$ is parallel to the edge of $R$ of length
$a_{i}$.

We state with out proof the following result.

######  Proposition 25.

Let $R$ be a rectangle in $\mathbb{R}^{n}$. Then there exists a smooth bump
function $\psi_{R}$ such that $\psi_{R}(x)\geq 1$ for $x\in R$ and $\psi_{R}$
decays rapidly outside $R$. And $\psi_{R}$ satisfies
$\mathrm{supp}(\mathcal{F}(\psi_{R}))\subset R^{*}$ and

(63) $|\mathcal{F}(\psi_{R})|\lesssim|R|\cdot 1_{R^{*}},$

Intuitively, we may think of $\psi_{R}$ as the indicator function $1_{R}$.

Figure 2.

Now, we are going to introduce the notion of rectangle and dual rectangle in
$\widetilde{A}(1,n)$, and obtain the inequality of form (63). We will view
$\widetilde{A}(1,n)$ as $\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$.

######  Definition 26.

We say $\mathbf{R}$ is a rectangle in $\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$
if $\mathbf{R}$ has the following form

(64) $\mathbf{R}=R_{0}\times R_{1}.$

Here, $R_{0}\subset\mathbb{R}^{n-1}$ is a rectangle and
$R_{1}\subset\mathbb{R}^{n-1}$ is a translated copy of $R_{0}$. We define the
dual rectangle to be

(65) $\mathbf{R}^{*}=R_{0}^{*}\times R_{1}^{*}.$

######  Remark 27.

See Figure 2. Since $R_{1}$ is a translated copy of $R_{0}$, we have
$R_{0}^{*}=R_{1}^{*}$. There are two ways to understand $\mathbf{R}$. On the
one hand, $\mathbf{R}$ is a Cartesian product of $R_{0},R_{1}$ as a subset of
$\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$. On the other hand, $\mathbf{R}$ is a
subset of $\widetilde{A}(1,n)$. If we use $P_{R_{0},R_{1}}$ to denote the
rectangle in $\mathbb{R}^{n}$ which is the convex hull of $R_{0}$ and $R_{1}$,
then $\mathbf{R}=\bigg{\\{}\ell\in\widetilde{A}(1,n):\ell\cap\\{0\leq
x_{n}\leq 1\\}\subset P_{R_{0},R_{1}}\bigg{\\}}$.

We have the following result.

######  Proposition 28.

Let $\mathbf{R}$ be a rectangle in $\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$.
Then there exists a smooth bump function $\psi_{\mathbf{R}}$ such that
$\psi_{\mathbf{R}}(X,Y)\geq 1$ for $(X,Y)\in\mathbf{R}$ and
$\psi_{\mathbf{R}}$ decays rapidly outside $\mathbf{R}$. And
$\psi_{\mathbf{R}}$ satisfies
$\mathrm{supp}(\widehat{\psi}_{\mathbf{R}})\subset\mathbf{R}^{*}$ and

(66) $|\widehat{\psi}_{\mathbf{R}}|\lesssim|\mathbf{R}|\cdot
1_{\mathbf{R}^{*}},$

Intuitively, we may think of $\psi_{\mathbf{R}}$ as the indicator function
$1_{\mathbf{R}}$.

###  5.2. Discretized estimate

We state a discretized version of (19). The setup is basically the same as
Theorem 21. Instead of (31), we have (69).

######  Theorem 29.

Fix a number $0<\mu<1/100$, and let $A_{\mu},G_{\mu}$ be as in Theorem 6. Fix
$t>0$, $0<s<a$. For sufficiently small $\varepsilon>0$ (depending on
$\mu,n,k,a,s,t$), the following holds. Let $0<\delta<1/100$. Let
$\mathbf{H}\subset A_{\mu}$ be a $(\delta,a)$-set with
$\\#\mathbf{H}\gtrsim(\log\delta^{-1})^{-2}\delta^{-a}$. Let
$\mathcal{V}\subset G_{\mu}$ be a $(\delta,t)$-set with
$\\#\mathcal{V}\gtrsim(\log\delta^{-1})^{-2}\delta^{-t}$. Suppose for each
$V\in\mathcal{V}$, we have a collection of slabs $\mathbb{T}_{V}$, where each
$T\in\mathbb{T}_{V}$ has dimensions
$\underbrace{\delta\times\dots\times\delta}_{(k-1)\
\textup{times}}\times\underbrace{1\times\dots\times 1}_{(n-k+1)\
\textup{times}}$ and is orthogonal to $V$ at some $\delta$-tube in $V$.

The $s$-dimensional Forstman condition holds for $\mathbb{T}_{V}$: for any
$\delta\leq r\leq 1$ and any $(n-k+1)$-dimensional $r$-slab $W_{r}$ that is
orthogonal to $V$ at some $r$-tube in $V$, we have

(67) $\\#\\{T\in\mathbb{T}_{V}:T\subset W_{r}\\}\lesssim(r/\delta)^{s}.$

We also assume that for each $\ell\in\mathbf{H}$,

(68) $\\#\\{V\in\mathcal{V}:\ell_{\delta}\subset
T,\textup{~{}for~{}some~{}}T\in\mathbb{T}_{V}\\}\gtrsim(\log\delta^{-1})^{-2}\\#\mathcal{V}.$

Then, we have

(69) $\delta^{-t-a}\lesssim_{\varepsilon}\delta^{-\varepsilon-(k+1)(n-k)-s}.$

The proof of (19) is deduced from Theorem 29 by a similar argument. We just do
not repeat here. We will focus on the proof of Theorem 29.

###### Proof of Theorem 29.

The idea is to use the Fourier transform on affine Grassmannian together with
the high-low method.

We set up some notation. Let $\mathbf{H}_{\delta}$ be the set of
$\delta$-balls in $A_{\text{loc}}(1,n)$ whose centers are points in
$\mathbf{H}$. We use $\ell$ to denote the elements in $\mathbf{H}$. By the
heuristic (23), we just denote the $\delta$-ball in $\mathbf{H}_{\delta}$ by
the corresponding bold-font

$\boldsymbol{\ell}_{\delta}:=\\{\ell^{\prime}\in
A_{\text{loc}}(1,n):d(\ell,\ell^{\prime})\leq\delta\\}.$

By (67), for each $V\in\mathcal{V}$, we have $s$-dimensional set of slabs
$\mathbb{T}_{V}=\\{T\\}$. For a slab $T$, we use the bold-font $\mathbf{T}$ to
denote a subset of $\widetilde{A}(1,n)$, so that each $\ell\in\mathbf{T}$
satisfies $\ell\cap B^{n}(0,1)\subset T$. Again, here we use the heuristic
(23).

Consider the the following integral

(70)
$\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}})^{2}.$

First of all, by (68) we notice that each
$\ell_{\delta}\in\mathbf{H}_{\delta}$ is contained in
$\gtrsim(\log\delta^{-1})^{-2}\delta^{-t}$ different $\mathbf{T}$. The volume
of a $\delta$-ball in $A(1,n)$ is $\sim\delta^{2(n-1)}$, therefore the volume
of $\mathbf{H}_{\delta}$ is $\sim\delta^{2(n-1)}\\#\mathbf{H}$ We have the
lower bound

(71)
$\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}})^{2}\gtrsim(\log\delta^{-1})^{-2t}\delta^{2(n-1)}\\#\mathbf{H}\delta^{-2t}\gtrsim_{\varepsilon}\delta^{\varepsilon}\delta^{2(n-1)}\delta^{-a}\delta^{-2t}.$

Our next goal is to obtain an upper bound for the integral. Recalling
Definition 26, since each $\mathbf{T}$ is a rectangle in
$A_{\text{loc}}(1,n)$, we write $\mathbf{T}=R_{0,\mathbf{T}}\times
R_{1,\mathbf{T}}$. Here, $R_{0,\mathbf{T}}$ is a rectangle in
$\mathbb{R}^{n-1}\times\\{0\\}$ of dimensions
$\underbrace{\delta\times\dots\times\delta}_{k-1\
\text{times}}\times\underbrace{1\times\dots\times 1}_{n-k\ \text{times}}$,
$R_{1,\mathbf{T}}$ is a translated copy of $R_{0,\mathbf{T}}$ which lies in
$\mathbb{R}^{n-1}\times\\{1\\}$. We can choose a smooth bump function

(72)
$\psi_{\mathbf{T}}(X,Y)=\psi_{R_{0,\mathbf{T}}}(X)\psi_{R_{1,\mathbf{T}}}(Y)$

adapted to $\mathbf{T}$ so that
$\mathrm{supp}\widehat{\psi}_{\mathbf{T}}\subset\mathbf{T}^{*}=R_{0,\mathbf{T}}^{*}\times
R_{0,\mathbf{T}}^{*}$.

Define

$f_{V}=\sum_{T\in\mathbb{T}_{V}}\psi_{\mathbf{T}}\text{~{}and~{}}f=\sum_{V\in\mathcal{V}}f_{V}.$

We will do the high-low decomposition. Let $K=(\log\delta^{-1})^{O(1)}$, where
$O(1)$ is a large number to be determined later.

Since we can identify $\widetilde{A}(1,n)$ and
$\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$ through the coordinates in (60) and
(61), we will constantly jump between these two spaces in the latter
discussion.

Choose a function $\eta(X,Y)$ on
$\widetilde{A}(1,n)=\mathbb{R}^{n-1}\times\mathbb{R}^{-1}$, such that
$\eta(X,Y)$ is a smooth bump function adapted to
$B^{2(n-1)}(0,(K\delta)^{-1})$. We have the following high-low decomposition
for $f_{V}$:

$f_{V}=f_{V,\text{high}}+f_{V,\text{low}},$

where $\widehat{f}_{V,\text{low}}=\eta\widehat{f}_{V}$ and
$\widehat{f}_{V,\text{high}}=(1-\eta)\widehat{f}_{V}$. For each
$(X,Y)\in\mathbf{H}_{\delta}$, we have

(73) $(\log\delta^{-1})^{-2}\\#\mathcal{V}\lesssim
f(X,Y)\leq|f_{\text{high}}(X,Y)|+|f_{\text{low}}(X,Y)|.$

We also let

(74) $f_{\textup{high}}=\sum_{V\in\mathcal{V}}f_{V,\textup{high}},\ \ \
f_{\textup{low}}=\sum_{V\in\mathcal{V}}f_{V,\textup{low}}.$

We will show that the high part dominates for $(X,Y)\in\mathbf{H}_{\delta}$,
i.e., $|f_{\text{high}}(X,Y)|\gtrsim(\log\delta^{-1})^{-2}\\#\mathcal{V}$. It
suffices to show

(75) $|f_{\text{low}}(X,Y)|\leq C^{-1}(\log\delta^{-1})^{-2}\\#\mathcal{V},$

for a large constant $C$.

Recall that $f_{\text{low}}=\sum_{V\in\mathcal{V}}f_{V}*\eta^{\vee}$. By the
definition of $\eta$, we see that $\eta^{\vee}$ is an $L^{1}$-normalized bump
function essentially supported in $B^{2(n-1)}(0,K\delta)$ and decays rapidly
out side of it. Let $\chi(X)$ be a positive function $=1$ on
$B^{n-1}(0,K\delta)$ and decays rapidly outside $B^{n-1}(0,K\delta)$. We have

(76) $|\eta^{\vee}(X,Y)|\lesssim(K\delta)^{-2(n-1)}\chi(X)\chi(Y).$

Together with (72), we have

(77)
$\begin{split}|f_{\text{low}}(X,Y)|&\lesssim\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}\psi_{\mathbf{T}}*\eta^{\vee}(X,Y)\\\
&\lesssim(K\delta)^{-2(n-1)}\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}\psi_{R_{0,\mathbf{T}}}*\chi(X)\psi_{R_{1,\mathbf{T}}}*\chi(Y)\end{split}$

Let $KR_{0,\mathbf{T}}$ be a tube of dimensions
$\underbrace{K\delta\times\dots\times K\delta}_{k-1\
\text{times}}\times\underbrace{1\times\dots\times 1}_{n-k\ \text{times}}$,
which is the $K$-dilation of the short edges of $R_{0,\mathbf{T}}$. Define
$KR_{1,\mathbf{T}}$ similarly. Let $K\mathbf{T}=KR_{0,\mathbf{T}}\times
KR_{1,\mathbf{T}}$. We see that $\psi_{R_{0,\mathbf{T}}}*\chi(X)$ is morally a
bump function at $KR_{0,\mathbf{T}}$ with weight $(K\delta)^{n-1}K^{-(k-1)}$.
Let us just ignore the rapidly decaying tail and write

(78)
$\psi_{R_{0,\mathbf{T}}}*\chi(X)\lesssim(K\delta)^{n-1}K^{-(k-1)}1_{KR_{0,\mathbf{T}}}(X).$

Similarly, we have

(79)
$\psi_{R_{1,\mathbf{T}}}*\chi(X)\lesssim(K\delta)^{n-1}K^{-(k-1)}1_{KR_{1,\mathbf{T}}}(X).$

Plugging back to (77), we obtain

(80) $|f_{\text{low}}(X,Y)|\lesssim
K^{-2(k-1)}\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{K\mathbf{T}}(X,Y).$

Fix $\ell=\ell(X,Y)\in\mathbf{H}$, let $W_{K\delta}$ be the $K\delta$-slab
that is orthogonal to $V$ at $P_{V}(\ell_{K\delta})$. By the $s$-dimensional
condition for $\mathbb{T}_{V}$, we have

(81)
$\sum_{T\in\mathbb{T}_{V}}1_{K\mathbf{T}}(X,Y)\lesssim\\#\\{T\in\mathbb{T}_{V}:T\subset
W_{K\delta}\\}\lesssim K^{s}.$

Plugging back to (77), we get

(82) $|f_{\text{low}}(X,Y)|\lesssim K^{s-2(k-1)}\\#\mathcal{V}.$

Noting that $s<2(k-1)$, we may choose $K\sim(\log\delta^{-1})^{O(1)}$ for
large $O(1)$ so that (75) holds.

Next, we want to regroup the slabs in $\mathbb{T}_{V}$. Note that each
$T\in\mathbb{T}_{V}$ is an $(n-k+1)$-dimensional $\delta$-slab, by the
transversality assumption, its intersection with
$\mathbb{R}^{n-1}\times\\{0\\}$ and $\mathbb{R}^{n-1}\times\\{1\\}$ are two
congruent $(n-k)$-dimensional $\delta$-slabs. We define $\mathbb{G}$ to be a
maximal $\delta$-separated subset of $G(n-k,n-1)$. We are going to define
$\mathbb{T}_{V}^{W}$ where $W$ ranges over $\mathbb{G}$. For each
$T\in\mathbb{T}_{V}$, we put $T$ into $\mathbb{T}_{V}^{W}$ if
$T\cap(\mathbb{R}^{n-1}\times\\{0\\})$ is parallel to $W$ up to
$\delta$-error. We obtain a partition

(83) $\mathbb{T}_{V}=\bigsqcup_{W\in\mathbb{G}}\mathbb{T}_{V}^{W}.$

We have the following observation. For
$T_{1}\in\mathbb{T}_{V_{1}}^{W},T_{2}\in\mathbb{T}_{V_{2}}^{W}$, we have that
if $T_{1}$ and $T_{2}$ are not comparable, then
$\mathbf{T}_{1}\cap\mathbf{T}_{2}=\emptyset$. The reason is that if $T_{1}$
and $T_{2}$ are not comparable, then either
$T_{1}\cap(\mathbb{R}^{n-1}\times\\{0\\})$ and
$T_{2}\cap(\mathbb{R}^{n-1}\times\\{0\\})$ are disjoint or
$T_{1}\cap(\mathbb{R}^{n-1}\times\\{1\\})$ and
$T_{2}\cap(\mathbb{R}^{n-1}\times\\{1\\})$ are disjoint, which means
$\mathbf{T}_{1}\cap\mathbf{T}_{2}=\emptyset$.

For $T\in\mathbb{T}_{V}^{W}$, we have

(84) $\mathrm{supp}(\widehat{\psi_{\mathbf{T}}})\subset W_{\delta}^{*}\times
W_{\delta}^{*}.$

Here, $W_{\delta}^{*}$ is the dual rectangle of $W_{\delta}$ in
$\mathbb{R}^{n-1}$. Therefore, we see that $W_{\delta}^{*}$ has dimensions
$\underbrace{\delta^{-1}\times\dots\times\delta^{-1}}_{k-1\
\text{times}}\times\underbrace{1\times\dots\times 1}_{n-k\ \text{times}}$.

Now, we have

(85)
$\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}})^{2}\lesssim\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}\psi_{\mathbf{T}})^{2}\lesssim\int_{\mathbf{H}_{\delta}}|f_{\text{high}}|^{2}\lesssim\int_{\widetilde{A}(1,n)}|f_{\text{high}}|^{2}.$

By Plancherel, it is further bounded by

(86)
$\int_{\widetilde{A}(1,n)}|\sum_{V\in\mathcal{V}}\sum_{W\in\mathbb{G}}\widehat{f_{V}^{W}}(1-\eta)|^{2}=\int_{\widetilde{A}(1,n)}|\sum_{W\in\mathbb{G}}\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}(1-\eta)|^{2}.$

We will estimate the overlap of
$\\{\mathrm{supp}(\sum_{V\in\mathcal{V}}\widehat{f}_{V}^{W}(1-\eta))\\}_{W\in\mathbb{G}}$.
We notice that

$\mathrm{supp}(\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}(1-\eta))\subset\bigg{(}W_{\delta}^{*}\times
W_{\delta}^{*}\bigg{)}\setminus B^{2(n-1)}(0,(K\delta)^{-1}).$

We pick $(\Xi_{1},\Xi_{2})\in\widetilde{A}(1,n)$ in the frequency space. We
assume $(\Xi_{1},\Xi_{2})$ lies in some
$\mathrm{supp}(\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}(1-\eta))$. Therefore,
we can assume without loss of generality that $\Xi_{1}\notin
B^{n-1}(0,\frac{1}{2}(K\delta)^{-1})$. For any $W\in\mathbb{G}$, if
$(\Xi_{1},\Xi_{2})\in\mathrm{supp}(\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}(1-\eta))$,
then $\Xi_{1}\in W^{*}_{\delta}$. Therefore, the overlap of
$\\{\mathrm{supp}(\sum_{V\in\mathcal{V}}\widehat{f}_{V}^{W}(1-\eta))\\}_{W\in\mathbb{G}}$
at $(\Xi_{1},\Xi_{2})$ is bounded by

(87) $\sum_{W\in\mathbb{G}}1_{W_{\delta}^{*}}(\Xi_{1}),$

Since $\Xi_{1}\notin B^{n-1}(0,\frac{1}{2}(K\delta)^{-1})$, we can further
bound (87) by the overlapping number of

(88) $\bigg{\\{}W_{\delta}^{*}\setminus
B^{n-1}(0,\frac{1}{2}(K\delta)^{-1})\bigg{\\}}_{W\in\mathbb{G}}.$

If we do a dilation by the factor $\delta$, then $W_{\delta}^{*}$ becomes
$(W^{\perp})_{\delta}$. So we just need to bound the overlapping number of

(89) $\bigg{\\{}(W^{\perp})_{\delta}\setminus
B^{n-1}(0,\frac{1}{2}K^{-1})\bigg{\\}}_{W\in\mathbb{G}}.$

We observe that when $W$ ranges over $\mathbb{G}$, $W^{\perp}$ will range over
$\delta$-separated subset of $G(k-1,n-1)$. By [1, Lemma 18], we see that the
overlapping number of (89) is bounded by

(90) $K^{O(1)}\delta^{-\dim(G(k-2,n-2))}$

The RHS of (86) is bounded by

(91)
$\begin{split}&K^{O(1)}\delta^{-\dim(G(k-2,n-2))}\int_{\widetilde{A}(1,n)}\sum_{W\in\mathbb{G}}|\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}(1-\eta)|^{2}\\\
\leq&K^{O(1)}\delta^{-\dim(G(k-2,n-2))}\int_{\widetilde{A}(1,n)}\sum_{W\in\mathbb{G}}|\sum_{V\in\mathcal{V}}\widehat{f_{V}^{W}}|^{2}\\\
=&K^{O(1)}\delta^{-\dim(G(k-2,n-2))}\sum_{W\in\mathbb{G}}\int_{\widetilde{A}(1,n)}|\sum_{V\in\mathcal{V}}f_{V}^{W}|^{2}\end{split}$

The last step is by Plancherel to return back to the physical space.

Now, for fixed $W\in\mathbb{G}$, we estimate

(92)
$\int_{\widetilde{A}(1,n)}|\sum_{V\in\mathcal{V}}f_{V}^{W}|^{2}=\int_{\widetilde{A}(1,n)}|\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}^{W}}\psi_{\mathbf{T}}|^{2}.$

We may ignore the rapidly decaying tail of $\psi_{\mathbf{T}}$ and think of it
as $1_{\mathbf{T}}$. To estimate the RHS of (92), we will estimate the
overlapping number of
$\\{\sum_{T\in\mathbb{T}_{V}^{W}}1_{\mathbf{T}}\\}_{V\in\mathcal{V}}$.

By an observation we addressed in the paragraph below (83), for any two tubes
$T_{1}\in\mathbb{T}_{V_{1}}^{W},T_{2}\in\mathbb{T}_{V_{2}}^{W}$, we have
either $T_{1}$ and $T_{2}$ are comparable, or $\mathbf{T}_{1}$ and
$\mathbf{T}_{2}$ are disjoint. Our task is as follows. For a given $T$, how
many $V\in\mathcal{V}$ can there be so that $T$ is comparable to some
$T^{\prime}\in\mathbb{T}_{V}^{W}$? Noting that if $T$ is comparable to some
element in $\mathbb{T}_{V}^{W}$, then $T$ must be morally orthogonal to $V$ at
some $\delta$-tube. Finding such $V$ is equivalent to finding
$\delta$-separated lines in $G(1,T)$. Therefore, for fixed $T$, the number of
such $V\in\mathcal{V}$ is $\lesssim\delta^{-(\dim T-1)}=\delta^{-(n-k)}$. As a
result, we obtain

(93)
$\int_{\widetilde{A}(1,n)}|\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}^{W}}1_{\mathbf{T}}|^{2}\lesssim\delta^{-(n-k)}\int_{\widetilde{A}(1,n)}\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}^{W}}1_{\mathbf{T}}.$

Combining (85),(86),(91),(92) and (93), we get the upper bound

(94)
$\begin{split}\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{T})^{2}\lesssim
K^{O(1)}\delta^{-(k-2)(n-k)}\delta^{-(n-k)}\int_{\widetilde{A}(l,n)}\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}}.\end{split}$

We notice that for any $(n-k+1)$-dimensional slab $S$, the dimension of

$\\{\ell\in\widetilde{A}(1,n):\ell\cap B^{n}(0,1)\subset S\\}$

is $2(n-k)$. If $S$ is the core of the $\delta$-slab $T$, or in other words
$T=S_{\delta}$, then

$\mathbf{T}=\\{\ell\in\widetilde{A}(1,n):\ell\cap B^{n}(0,1)\subset T\\}$

is roughly the $\delta$-neighborhood of $\\{\ell\in\widetilde{A}(1,n):\ell\cap
B^{n}(0,1)\subset S\\}$ in $\widetilde{A}(1,n)$, which has measure
$\sim\delta^{2(n-1)-2(n-k)}$. We also note that
$K\sim(\log\delta^{-1})^{O(1)}$. We can bound the RHS of (94) by

(95)
$\lesssim_{\varepsilon}\delta^{-\varepsilon}\delta^{-(k-2)(n-k)}\delta^{-(n-k)}\delta^{-s-t}\delta^{2(n-1)-2(n-k)}$

In summary, we obtain

(96)
$\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}})^{2}\lesssim_{\varepsilon}\delta^{-\varepsilon}\delta^{-(k-2)(n-k)}\delta^{-(n-k)}\delta^{-s-t}\delta^{2(n-1)-2(n-k)}.$

Comparing with the lower bound

(97)
$\int_{\mathbf{H}_{\delta}}(\sum_{V\in\mathcal{V}}\sum_{T\in\mathbb{T}_{V}}1_{\mathbf{T}})^{2}\gtrsim\delta^{2(n-1)}\\#\mathbf{H}\delta^{-2t}\sim\delta^{2(n-1)}\delta^{-a}\delta^{-2t},$

we obtain

(98) $t\leq k(n-k)+s-a+(n-k).$

∎

##  6\. The projection problem for $A(l,n)$

It is quite natural to further think about the same projection problem for
$A(l,n)$. We formally state the problem.

Fix integers $0<l<k<n$. Let $V\in G(k,n)$. Note that for $L\in A(l,n)$, we
have $P_{V}(L)\in A(j,V)$ for some $0\leq j\leq k$. We can define

$\pi_{l,V}:A(l,n)\rightarrow\bigsqcup_{j=0}^{l}A(j,V)$ $L\mapsto P_{V}(L).$

We can ask the same Marstrand-type projection problem. For
$0<a<(l+1)(n-l)=\dim(A(l,n))$, what is the optimal number $s(n,k,l,a)$ such
that the following is true? Let $\mathbf{A}\subset A(l,n)$ with
$\dim(\mathbf{A})=a$. Then we have

(99) $\dim(\pi_{l,V}(\mathbf{A}))=s(n,k,l,a),\ \textup{for~{}a.e.~{}}V\in
G(k,n).$

For simplicity, we assume $n,k,l$ are fixed. So we write $\pi_{V}$ for
$\pi_{l,V}$.

######  Definition 30.

Fix $0\leq l<k<n$. For any $0<a<\dim(A(l,n))$, define

(100) $S(a):=\inf_{\mathbf{A}\subset
A(l,n),\dim(\mathbf{A})=a}\operatorname*{ess\,sup}_{V\in
G(k,n)}\dim(\pi_{V}(\mathbf{A})).$

Here, we require $\mathbf{A}$ to be a Borel set to avoid some measurability
issue. We also remark that $S(a)=S_{l,k,n}(a)$ should also depend on $l,k,n$,
but we just omit them from the notation for simplicity.

A reasonable conjecture would be:

######  Conjecture 31.

For $j=0,1,\dots,l$, we have the following lower bounds of $S(a)$.

(101) $\begin{split}S(a)=a-(l-j)(n-k)\ \ \ a\in[(l-j)(n-l),(l-j)(n-l)+k-l].\\\
S(a)=(l-j+1)(k-l)\ \ \ a\in[(l-j)(n-l)+k-l,(l-j+1)(n-l)].\end{split}$

We are able to show the upper bound of $S(a)$ by constructing examples as
follows.

######  Proposition 32.

For $j=0,1,\dots,l$, we have the following lower bounds of $S(a)$.

(102) $\begin{split}S(a)\leq a-(l-j)(n-k)\ \ \
a\in[(l-j)(n-l),(l-j)(n-l)+k-l].\\\ S(a)\leq(l-j+1)(k-l)\ \ \
a\in[(l-j)(n-l)+k-l,(l-j+1)(n-l)].\end{split}$

###### Proof.

Fix $j\in\\{0,\dots,l\\}$. Let $V_{j}=\mathbb{R}^{j}\times\\{0\\}^{n-j}$,
$V_{l}=\mathbb{R}^{l}\times\\{0\\}^{n-l}$,
$V_{k}=\mathbb{R}^{k}\times\\{0\\}^{n-k}$. We have $V_{j}\subset V_{l}\subset
V_{k}\subset\mathbb{R}^{n}$. We introduce a new notation. For $j<l$ and $V\in
G(j,n)$, define

(103) $\textup{Bush}(l,V):=\\{W\in G(l,n):W\supset V\\},$

which is the set of $l$-subspaces that contain $V$. We call $V$ the stem of
the bush. It is not hard to see for $V\in G(j,n)$,

(104) $\dim(\textup{Bush}(l,V))=(l-j)(n-l).$

We first consider the case $a=(l-j)(n-l)+b$ where $0\leq b\leq k-l$. We choose
$B\subset V_{l}^{\perp}$, so that $\dim(B)=b$. Note that this is allowable
since $\dim(V_{l}^{\perp})=n-l\geq b$. Next, we choose

(105) $\mathbf{A}=\bigcup_{v\in B}\textup{Bush}(l,V_{j}+v).$

One way to think about $\mathbf{A}$ is as follows. We first locate the set of
stems $\\{V_{j}+v\\}_{v\in B}$, and then enrich each stem to become a bush
$\textup{Bush}(l,V_{j}+v)$. $\mathbf{A}$ is the union of these bushes.

We compute the dimension of $\mathbf{A}$. The idea is that we have
$b$-dimensional set of bushes and each bush has dimension $(l-j)(n-l)$, so we
have

$\dim\mathbf{A}\leq\dim B+\dim(\textup{Bush}(l,V_{j}))=(l-j)(n-l)+b.$

The equality holds if there is not too much overlap between different bushes.
We handle it in the following way. Consider

(106)
$\textup{Bush}^{\prime}(l,V_{j}+v):=\\{W\in\textup{Bush}(l,V_{j}+v):W\cap
V_{l}^{\perp}=\\{0\\}\\},$

which are the $l$-planes in the bush that are transverse to $V_{l}^{\perp}$.
It is not hard to see

$\dim(\textup{Bush}^{\prime}(l,V_{j}+v))=\dim(\textup{Bush}(l,V_{j}+v))=(l-j)(n-l).$

We also have that for $v_{1}\neq v_{2}\in B$,

(107)
$\textup{Bush}^{\prime}(l,V_{j}+v_{1})\cap\textup{Bush}^{\prime}(l,V_{j}+v_{2})=\emptyset.$

We prove it. Suppose there exists
$W\in\textup{Bush}^{\prime}(l,V_{j}+v_{1})\cap\textup{Bush}^{\prime}(l,V_{j}+v_{2})$.
Then $W\supset V_{j}+v_{1},V_{j}+v_{2}$ which implies $W\supset v_{1}-v_{2}$.
However $v_{1}-v_{2}\in V_{l}^{\perp}$ which contradicts $W\cap
V_{l}^{\perp}=\\{0\\}$.

Now we have

(108) $\mathbf{A}\supset\bigsqcup_{v\in B}\textup{Bush}^{\prime}(l,V_{j}+v),$

so $\dim(\mathbf{A})\geq\dim
B+\dim(\textup{Bush}^{\prime}(l,V_{j}+v))=(l-j)(n-l)+b$. Therefore, we have

(109) $\dim(\mathbf{A})=(l-j)(n-l)+b=a.$

Next, we show that for generic $V\in G(k,n)$,

(110) $\dim(\pi_{V}(\mathbf{A}))\leq(l-j)(k-l)+b=a-(l-j)(n-k).$

We need the following notation. For subspaces $W_{1}\subset W_{2}$ with $\dim
W_{1}\leq i\leq\dim W_{2}$, define

(111) $\textup{Bush}(l,W_{1},W_{2}):=\\{W\in G(l,W_{2}):W\supset W_{1}\\}.$

One can check that
$\textup{Bush}(l,W_{1},\mathbb{R}^{n})=\textup{Bush}(l,W_{1})$ and
$\textup{Bush}(l,\\{0\\},W_{2})=A(l,W_{2})$.

We prove (110). If $V\in G(k,n)$ satisfies $V_{j}\cap V^{\perp}=\\{0\\}$ (or
in other words, $\pi_{V}(V_{j})$ is $j$-dimensional), then for each bush
$\textup{Bush}(l,V_{j}+v)$ in $\mathbf{A}$, we have

(112)
$\pi_{V}(\textup{Bush}(l,V_{j}+v))=\textup{Bush}(l,\pi_{V}(V_{j}+v),V)\sqcup\bigsqcup_{j\leq
i\leq l-1}\textup{Bush}(i,\pi_{V}(V_{j}+v),V).$

In other words, $\pi_{V}$ project the bush $\textup{Bush}(l,V_{j}+v)$ to the
set of planes of dimension less than $l$ in $V$ that contain
$\pi_{V}(V_{j}+v)$. Since the first term on the right hand side of (112)
dominates the Hausdorff dimension, we have

(113)
$\dim(\pi_{V}(\textup{Bush}(l,V_{j}+v)))=\dim(\textup{Bush}(l,\mathbb{R}^{j},\mathbb{R}^{k}))=(l-j)(k-l).$

Therefore,

(114) $\dim(\pi_{V}(\mathbf{A}))\leq\dim B+(l-j)(k-l)=b+(l-j)(k-l),$

which finishes the proof of (110), and hence the first part of (102).

We prove the second part of (102). We just need to show that for
$a=(l-j+1)(n-l)$. The idea is similar to (105), but here we choose
$B=V_{l}^{\perp}$. We construct

(115) $\mathbf{A}=\bigcup_{v\in V_{l}^{\perp}}\textup{Bush}(l,V_{j}+v).$

By the same reasoning, we have

$\dim(\mathbf{A})=\dim(V_{l}^{\perp})+(l-j)(n-l)=(l-j+1)(n-l).$

On the other hand, for generic $V\in G(k,n)$, $\pi_{V}(\mathbf{A})$ consists
of $(k-l)$-dimensional set of bushes, since the stems of these projected
bushes are parallel $l$-planes in $V$. Also, the dimension of each bush is
$(l-j)(k-l)$. Therefore, we have

(116) $\dim(\pi_{V}(\mathbf{A}))\leq(k-l)+(l-j)(k-l)=(l-j+1)(k-l).$

This finishes the second part of (102).

∎

However, with the current method in the paper, we are not able to show the
lower bound of $S(a)$ for all range of $a$. We can also prove Falconer-type
and Kaufman-type exceptional set estimates, and obtain some partial results.
We just state it without proof.

######  Theorem 33.

For $j=0$ or $l$, we have the following lower bound of $S(a)$.

(117) $\displaystyle S(a)\geq a-(l-j)(n-k)\ \ \
a\in[(l-j)(n-l),(l-j)(n-l)+k-l].$ (118) $\displaystyle S(a)\geq(l-j+1)(k-l)\ \
\ a\in[(l-j)(n-l)+k-l,(l-j+1)(n-l)].$

$j=0$ corresponds to the Falconer-type estimate, while $j=l$ corresponds to
the Kaufman-type estimate.

## References

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# Equilivest: A Robotic Vest to aid in Post-Stroke Dynamic Balance
Rehabilitation

Franco Paviotti1, Esteban Buniak 2, Rodrigo Ramele 3, Orestes Freixes4 and
Juan Miguel Santos5 1F. Paviotti, 2E.Buniak and 3R.Ramele are with the
Instituto Tecnológico de Buenos Aires (ITBA) Buenos Aires, Argentina
<EMAIL_ADDRESS>Freixes is with the CINER Centro
Integral de Neurorehabilitación, Buenos Aires, Argentina
<EMAIL_ADDRESS>Santos is with the UNAHUR Universidad Nacional
de Hurlingham, Buenos Aires, Argentina<EMAIL_ADDRESS>extended
abstract was presented at the Workshop on Assistive Robotic Systems for Human
Balancing and Walking: Emerging Trends and Perspectives at IROS2022.

###### Index Terms:

Stroke, Dynamic Balance, Neurorehabilitation, Biofeedback, Vibrotactile

##  I Introduction

Brain stroke is a devastating medical condition, that affects world population
and is the main cause of disabilities worldwide [1]. Disabilities related to
stroke can affect motor pathways, and may lead to several motor function
disorders. One important aspect of motor function is balance which is the
ability to control the body’s center of mass inside the base support provided
by the lower limb [2]. Stroke can affect dynamic balance as well, which is
manifested while walking, impairing autonomy and independence, important
factors in Activities of Daily Living (ADL) particularly for young patients
[3, 4].

Strong evidence suggests that neuroplasticity can be enhanced by neural
rehabilitation [5, 6]. These procedures are aimed to relearn movements that
can trigger new neural pathway generation which reroute, or even completely
replace, those pathways that were damaged by the stroke. Recently, biofeedback
techniques aiming to provide extra information to the patient to aid in the
relearning, have appeared as a complementary treatment to increase
neuralplasticity. These are in the form of Wearable devices-based biofeedback
rehabilitation (WDBR) [7] or robotic rehabilitation gait devices [8, 9, 10].

Therefore, the addition of an independent and new peripheral therapeutic
signal, that can be assimilated as an extrasensory input [11], could improve
dynamic balance performance on post-stroke patients which may have yet
insufficiency to deal properly with the complexities of walking. Meaningful
balance information, in terms of timing and location, can provide this extra
signal in any form of stimulation, particularly vibrotactile feedback (VF)
[12]. Although the effectiveness of biofeedback on static balance has been
studied more extensively in the literature [5], to the best of our knowledge
works dealing with dynamic balance problems while walking have been more
scarce.

This work presents the development of a device which is grounded on this idea,
and aims to help a post-stroke patient with a remaining dynamic balance
problem, presenting it as a case study. The proposed development is
implemented as a smart-vest [11], which we will call, Equilivest, that address
three possible clinical hypothesis of the underlying problem. We aim to
provide motor learning, meaning to generate an assist-as-needed [13, 14]
vibrotactile feedback signal which initially promotes voluntary control over
movements of the joints [12], but which fades away to encourage automatism
[15, 4]. The device aims to increase plasticity by producing timing
vibrotactile stimulation based on kinematic and dynamics measurements.

##  II Materials and Methods

###  II-A Patient Case Study

Patient is a 36 years-old female, without any history of chronic ailment, who
suffered an acute brainstem stroke after giving birth. The stroke was on
posterior fossa subarachnoid due to a brain artereovenous malformation (AVM),
which could be linked to pregnancy or puerperium [16]. Patient was in induced
coma for around 1 month, and after that unable to walk, move, talk or swallow.
Throughout a first period of intensive rehabilitation, the patient managed to
recover significantly, including from dysphagia.

After 7 months since event, the patient was discharged from hospital and
maintained a daily rehabilitation treatment, focusing on a remaining affection
related to dynamic balance problems while walking. The patient achieved
satisfactory index scores in static balance tests and is able to perform
static hip-balance and ankle-balance. She has recovered muscle force in her
legs and can perform lower-limb exercises. Her vision is normal.

However, when the patient tries to walk on open-spaces, or with confronting
lights (like walking towards sunlight), with other people moving around, or
when concentration fades while walking, she is unable to keep up with the pace
of the gait and the risk of fall increases, in a non-lateralized locomotion
(symmetric) [17]. Nowadays, the patient walks freely unaided at home but
requires a Canadian cane otherwise and constant supervision.

###  II-B Underlying hypothesis

Human balance is composed of a complex interaction of different subsystems,
which includes somatosensorial information, vestibular system and visual
information as input sources. These are later processed in different networks
of the Central Nervous System (CNS), and finally actuated by motor pathways at
many different scales [4].

Question |  Answer
---|---
Under which situations do you feel you are prone to fall ? |  When I am tired, distracted or stressed.
What do you feel ? |  I feel that I loss my balance
What do you feel before you fall ? |  Sometimes I do feel something before. I realize that I am going to fall, but there isn’t anything I can do to avoid it.
When do you use the cane ? |  Only when I am out of home.
TABLE I: Survey and responses provided by the patient. Question |  Answer
---|---
Describe the situation when the patient loss her balance |  Loss of balance is manifested at the beginning, during the gait cycle or with sudden stops. Walking is performed using intense visual compensation. Instability increase when the patient is distracted, when she stops looking at the floor, or when she looks sideways. Her gait is not automated, and demands cognitive resources.
Does she presents any of the following muscular dystrophy in lower-limbs / dysmetria / heavy shaking / somatosensorial alterations / proprioceptive alterations ? |  No
Does she have any visible reaction before falling? |  She presents dynamic alterations in hip and ankle compensation.
Is the patient gait normal ? |  No, the patient presents ataxic gait, likely triggered by slow reaction to perform lower-limb balance correction.
Is there any behavioral pattern linked to falling events ? |  Yes, cognitive workload. Dual tasks situations, when the patient needs to perform something extra while walking.
Describe current treatment. |  The patient is currently working on rehabilitation exercises to retrain her Vestibulo-Ocular Reflex (VOR), to improve her static and dynamic balance, and improve her locomotor automatism. She is improving on a monthly basis, but still she has not reached the level to walk autonomy without any aid.
TABLE II: Survey and responses provided by patient’s rehabilitation caregiver.

We perform a series of surveys and interviews with the patient and their
caregivers. Main results are summarized in Tables I and II. Based on these
results and patient’s clinical history, we postulate three different potential
clinical situations that could use the external biofeedback signal eventually
aiding in rehabilitation procedures. The first (i) is a potential problem in
the integration of vestibular information while walking, the second (ii) is
bradykinesia where the required processing speed to effectively adjust the
lower limb to keep the center of mass inside the base support is not achieved.
Finally, the third (iii) hypothesis is an ataxic gait, where non-automated
gait produces an increase in the likelihood of failing.

###  II-C Robotic Device Vest

The vest system prototype is based on Internet of Robot Thing (IoRT)
technology [18, 19]. The controller is an Arduino UNO (Arduino LLC, Italy)
board coupled with ESP8266 shield (WeMos, United Kingdom). The system is
powered by a commercial power bank with Li-Po 18650 cells (Ipower, United
States). It also contains an Inertial Measurement Unit (IMU) MPU 6050 (OEM
ITG/MPU6050) and a FA-12350 DC motor scavenged from old compact discs which
provide the vibrotactile feedback. The IMU provides accelerometer and
gyroscope data to get a 3-dimensional angular acceleration vector and a linear
acceleration vector. Roll (Z), yaw (Y) and pitch (X) angles, aligned
respectively with anatomical planes coronal, horizontal and sagittal are
calculated from a set of transforming equations and processed with a
complementary filter which relies $0.98$ in gyroscope data and $0.02$ in
accelerometer data [20]. These values are transmitted by telemetry in real-
time as UDP packets for off-board register, processing and further analysis.
Data recovered by the device is sampled at 100 Hz. The smart-vest prototype
can be seen in Figure 1. It is based on a safety vest, complemented with
Velcro pockets.

Figure 1: Front and back views of the Equilivest prototype. The vest is based
on a safety vest, with additional velcro pockets. The battery is located on
the back, while the vibrotactile motor is located on the belly [11]. The IMU
sensor is pressed against the chest by the use of a flexible elastic band.

###  II-D Experimental Design

Three experiments are designed in order to assess each one of the potential
clinical causes, and to derive for each one of them a stimulation strategy.
Participants are recruited voluntarily and the experiment is conducted
anonymously in accordance with the declaration of Helsinki published by the
World Health Organization. No monetary compensation is handed out and all
participants agree and sign a written informed consent. This study is approved
by the Departamento de Investigación y Doctorado, Instituto Tecnológico de
Buenos Aires (ITBA). All the participants wear the smart-vest with an elastic
band pressed tightly towards the chest, tied in no-restrictive manner, holding
the sensor firmly.

####  II-D1 Vestibular Information Integration

This experiment aims to study the falling process. Participants are told to
lean forward performing ankle forward until they couldn’t maintain balance
anymore and let themselves fall into a mattress. This experiment is used to
determine a threshold for the data that can help to identify the breakpoint
conditions based on the IMU information where each subject falls.

In order to test it, 5 healthy participants are recruited to perform 10 runs
of falling situations Participants wearing the vest lean forward by adjusting
their ankle joint, without performing a one-step forward movement, at
different angles progressively until they can no longer cope with the
unbalance situation and fall to the mattress.

####  II-D2 Ataxic Gait

This experiment has the purpose of analyzing and studying gait’s behavior. In
order to accomplish this, 5 participants are instructed to execute 10 sessions
of walking in a straight line across ten meters, performing Ten-Meter Walking
(10MWT) assesment [21]. We collected and analyzed pitch, roll and yaw values
as well as angular acceleration changes registered by the IMU gyroscope. In
this study our goal is to be able to process and identify gait abnormalities
as well as all steps occurring and any fall that might develop.

####  II-D3 Bradykinesia

This experiment involves the coupling of the other two. Five participants are
instructed to perform 10 runs of a ten-meter walking procedure, followed by a
falling into a mattress. The purpose of this experiment is obtain a
multichannel time series of the whole sequence, including the walk and the
falling action.

## Results and Discussion

Preliminary results show that for the first (1) experiment, on Figure 2, the
pitch angle can be used to determine a breakpoint where the fall is inevitable
(black vertical line, marked as 1*). On the other hand, results from
experiment number (2) in Figure 3 show that the gait cycle is clearly visible
from raw acceleration data.

Figure 2: Pitch value in degrees and angular acceleration around X axis
showing the progressive increase as participants leaned forward crossing the
instability condition (1). Time is in milliseconds.  Figure 3: Angular
acceleration for the X and Z axis showing 7 steps of the 10MWT assessment.
Time is in milliseconds.

The information provided from all the experiments allow us to implement three
different VF strategies: The first is an Artificial Vestibular Feedback, which
provides information as a feedback for vestibular information integration
process. This is implemented as a frequency-increasing VF signal that
increases as long as the distance from the breaking condition position where
the fall is inevitable, decreases. The purpose is to provide the patient with
a continuous sensation that modulates the risk of falling.

The second strategy, independent of the first one, is the implementation of
Gait Pacemaker. The obtained data from the analysis of the ataxic gait is used
to implement a podometer. It has been showed that gait synchronization with
music increase gait performance [22] and synchronized gait trainer achieved
positive results on patient [23]. Hence, the aim of this stimulation approach
is to keep the patient as close as possible to a normal and safe gait.

The final stimulation scheme deals with the Bradykinesia condition, and aims
to have a Risk-Predictor [24, 25]. The IMU data represents a multichannel time
series. Hence, it can be used to predict a risk of falling, ahead of its
occurrence, by building a machine-learning predictor based on the analysis of
the time-series data. The rationale is to counter the slowness of the response
by predicting the falling situation ahead of its occurrence and providing the
VF stimulation to force a change in the risky gait pattern.

## Conclusion

Overall, we aim for Equilivest to serve as a testbed that can be iteratively
extended to obtain experimental data and to implement biofeedback strategies
[2].

## References

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