Sharp Fourier decay estimates for measures supported on the well-approximable numbers
Robert Fraser Thanh Nguyen


 Vacuum Radiation Pressure Fluctuations on Electrons
    L. H. Ford
    September 9, 2024
====================================================



§ ABSTRACT 
We construct a measure on the well-approximable numbers whose Fourier transform decays at a nearly optimal rate. This gives a logarithmic improvement on a previous construction of Kaufman.




§ INTRODUCTION AND BACKGROUND




 §.§ Harmonic analysis on fractal sets

An interesting class of problems in harmonic analysis involves determining information about the Fourier transform of a compactly supported measure μ given information about the support μ of the measure μ. A standard result in this area is Frostman's lemma, which states that if E is a set of Hausdorff dimension s, then for any t < s, there exists a Borel probability measure μ_t supported on E satisfying the condition that

    ∫_ξ∈ℝ^n |μ̂_t(ξ)|^2 (1 + |ξ|)^-t < ∞.

Frostman's lemma states that, up to an ϵ-loss in the exponent, the set E supports a measure whose Fourier transform decays like |ξ|^-s/2 in an L^2-average sense.

This version of Frostman's lemma motivates the definition of Fourier dimension. The Fourier dimension of a set E ⊂ℝ^n is the supremum of those values 0 ≤ s ≤ n such that E supports a Borel probability measure μ_s satisfying the pointwise condition

    |μ̂_s(ξ)| ≲ (1 + |ξ|)^-s/2.

Observe that the condition (<ref>) for some value of s implies equation (<ref>) for any t < s. However, there is no reason to expect a converse statement to hold; in fact, if E is the usual middle-thirds Cantor set, there is no Borel probability measure μ on E such that |μ̂(ξ)| → 0 as |ξ| →∞. A measure μ such that |μ̂(ξ)| → 0 as ξ→∞ is called a Rajchman measure. On the opposite extreme, there are a number of examples of sets E of Hausdorff dimension s supporting Borel probability measures satisfying (<ref>) for all t < s. Such sets are called Salem sets.

If s = n - 1, a simple stationary phase calculation shows that the usual surface measure on the sphere satisfies the condition

    |μ̂(ξ)| ≤ (1 + |ξ|)^-(n-1)/2.

This well-known computation can be found in the textbooks of Wolff <cit.> and Mattila <cit.>.
If n = 1 and 0 < s < 1, the first examples of Salem sets were given by Salem <cit.> via a random Cantor set construction. A later random construction was given by Kahane <cit.>, who shows that if Γ : [0,1] →ℝ^n is a Brownian motion and E ⊂ [0,1] is a set of Hausdorff dimension s, then Γ(E) will almost surely have Fourier dimension equal to 2s. Kahane <cit.> also constructed Salem sets using random Fourier series whose coefficients are given by Gaussian random variables.

The first explicit, deterministic example of a Salem set of fractional dimension in ℝ was given by Kaufman <cit.>. For an exponent τ, the well-approximable numbers E(q^-τ) are defined by

    E(q^-τ) = {x : | x - p/q| ≤ q^-τ for infinitely many pairs of integers (p,q)}.

A classical result of Jarník <cit.> and Besicovitch <cit.> states that the Hausdorff dimension of E(q^-τ) is equal to 2/τ. Kaufman shows that E(q^-τ) supports a Borel probability measure μ satisfying

    |μ̂(ξ)| ≲ (1 + |ξ|)^-1/τ o(log |ξ|).

Bluhm <cit.> provides an exposition of Kaufman's argument to prove a slightly weaker result in which the o(log |ξ|) term is replaced by O(log |ξ|).
More generally, given a function ψ : ℕ→ [0, ∞), it is of interest to consider the set of ψ-approximable numbers

    E(ψ) = {x : | x - p/q| ≤ψ(q)  for infinitely many pairs of integers (p,q)}.

Hambrook <cit.> obtains lower bounds on the Fourier dimension of such sets in terms of the function ψ.


 §.§ Some problems in geometric measure theory

In this paper, we will consider the question of locating sets E satisfying more precise estimates than (<ref>) under the constraint that E has finite Hausdorff measure. As a motivating example, consider the (n-1)-dimensional sphere in ℝ^n. This set has positive and finite (n-1)-dimensional Hausdorff measure and supports a measure μ with Fourier transform satisfying (<ref>). Mitsis <cit.> posed the following problem.


[Mitsis's problem]
For which values of 0 < s < n does there exist a measure μ such that μ simultaneously satisfies the ball condition

    μ(B(x,r)) ∼ r^s  for all x ∈μ and all r > 0

and the Fourier decay condition

    |μ̂(ξ)| ≤ |ξ|^-s/2?



We will consider a related problem. Let 0 ≤ s ≤ n. Recall that a subset E of ℝ^n is said to be an s-set if the Hausdorff measure ℋ^s(E) satisfies 0 < ℋ^s(E) < ∞.
[Fourier transform on s-sets]
For which values of 0 < s < n does there exist an s-set E supporting a measure μ such that μ satisfies the Fourier decay condition

    |μ̂(ξ)| ≤ |ξ|^-s/2?


Of course, such a set E must be a Salem set of Hausdorff dimension s.

This problem can be extended to a question about generalized Hausdorff dimension. Recall that a positive, increasing function α is said to be a dimension function if α(u) → 0 as u → 0. We will say that E is an α-set if 0 < ℋ_α(ℰ) < ∞, where ℋ_α is the generalized Hausdorff measure associated to α. The following question generalizes the previous one:
[Fourier transform on α-sets]
For which dimension functions α does there exist an α-set E supporting a measure μ such that μ satisfies the Fourier decay condition

    |μ̂(ξ)| ≲√(α(1/ξ))  for |ξ| ≥ 1?



We conjecture that the only such dimension functions α are integer powers α(u) = u^-s for integers 0 ≤ s ≤ n.

On the other hand, we also wish to pose the problem of determining the optimal Fourier decay estimates for measures supported on the set of well-approximable numbers E(ψ).
[Fourier decay of measures supported on E(ψ)]
Fix a function ψ. For which functions Θ does there exist a measure μ supported on E(ψ) such that

    |μ̂(ξ)| ≲Θ(ξ)?


Although we are unable to answer Problems <ref>, <ref>, and <ref> in this work, we are able to obtain “near"-answers to all three of these questions if the dimension function α or the approximation function ψ decay at a polynomial rate.


 §.§ Notation

In this paper, constants are always allowed to depend on the parameters τ, σ, and ρ. Any dependence on these parameters will always be suppressed for simplicity of notation.

If A and B are any two quantities, we write A = O(B) or A ≲ B to imply that A ≤ C B for some constant C that does not depend on A or B (but may depend on τ, σ, or ρ). We write B ≳ A to mean the same thing as A ≲ B. If A ≲ B and B ≲ A, we write A ∼ B. If the implicit constant in any of these inequalities is allowed to depend on some other parameter such as ϵ, we write A ≲_ϵ B, A ≳_ϵ B, or A ∼_ϵ B. 

If A(x) and B(x) are functions of a variable x, we write A(x) ⪅ B(x) if A(x) ≲_ϵx^ϵ B(x) for every ϵ > 0. So, for example, we write 

    x^3 exp (√(log x)) log x loglog x ⪅ x^3.

If A(x) ⪅ B(x) and B(x) ⪅ A(x), we write A(x) ≈ B(x).


§ RESULTS

First, we describe a result in the direction of Problem <ref>.
 
Let ψ(q) be an arbitrary nonnegative, decreasing function satisfying the conditions

    2 < lim_q →∞ -log(ψ(q))/log q = τ < ∞.

Suppose also that there exists σ > 1 such that ψ satisfies the polynomial-type decay condition

    ψ(q_1)/ψ(q_2)≥(q_2/q_1)^σ  for q_2 > q_1 sufficiently large.

Suppose further that 1 ≤χ(q) ≤log q is a nonnegative function that satisfies 

    ∑_q=1 
     q  prime^∞1/qχ(q) = ∞.

and also satisfies the subpolynomial-type growth condition for any ϵ > 0:

    χ(q_2)/χ(q_1) < (q_2/q_1)^ϵ  for q_1, q_2 sufficiently large depending on ϵ.


Then for any increasing function ω with lim_ξ→∞ω(ξ) = ∞, there exists a Borel probability measure μ supported on a compact subset of the ψ-well-approximable numbers satisfying the estimate

    |μ̂_χ, ω(ξ)| ≲ω(|ξ|)/ψ^-1(1/|ξ) χ (ψ^-1(1 /|ξ|))  for all ξ∈ℝ.


In order to simplify our notation, we define

    θ(ξ) := 1/ψ^-1(1/ξ) χ(ψ^-1(1/ξ)).


If ψ(q) = q^-τ, Theorem <ref> gives estimates that improve on those of Kaufman <cit.>. In this case, the estimate (<ref>) becomes

    |μ̂_χ, ω(ξ)| ≲ |ξ|^-1/τω(|ξ|)/χ(|ξ|^1/τ).

Observe that, for example, the choice χ(q) = loglog q satisfies (<ref>). On the other hand, ω can be taken to be any function that increases to ∞, so it is possible to choose ω(ξ) = logloglogξ, for example. Hence there exists a measure μ supported on the well-approximable numbers satisfying

    |μ̂(ξ)| ≲ |ξ|^-1/τlogloglog |ξ|/loglog |ξ|≪ |ξ|^-1/τ.




Our next result is in the direction of Problem <ref>.
 
Let α be a dimension function with

    0 < lim_x → 0logα (x)/log x = ν < ∞

and for some ρ < 1 such that 

    α(x_1)/α(x_2)≥(x_1/x_2)^ρ

for sufficiently small x_1 < x_2.

Let ω be an increasing function such that lim_ξ→∞ω(ξ) = ∞. Then there exists a compact set F_α of zero α-Hausdorff measure such that there exists a measure μ_α, ω supported on F_α satisfying

    |μ̂(ξ)|≲√(α (1/|ξ|))ω (|ξ|)

for all s ≠ 0. Such a set is given by an appropriately chosen subset of the well-approximable numbers E(ψ) where

    ψ(q) = α^-1(q^-2).



Although this does not provide an answer to Problem <ref>, it comes within an arbitrarily slowly growing function of answering this problem. In other words, any improvement on the estimate of Theorem <ref> will give an answer to Problem <ref>. 


Observe that the condition (<ref>) on α implies the condition (<ref>) on ψ for τ = 2/ν. A simple calculation also shows that the condition (<ref>) implies the condition (<ref>) with σ = 2/ρ. This is the only way in which the assumptions (<ref>) and (<ref>) will be used.

Finally, we show that, for any decreasing approximation function ψ, the set E(ψ) supports a Rajchman measure. This improves a result of Bluhm <cit.> constructing a Rajchman measure supported on the set of Liouville numbers.
 
For an arbitrary nonnegative, decreasing function ψ there exists a Rajchman measure, μ, supported on a compact subset of the ψ-well-approximable numbers.

In a recent work, Polasek and Rela <cit.> improve Bluhm's result in a different way by showing an explicit Fourier decay bound on the set of Liouville numbers. They show that if f : ℝ^+ →ℝ^+ is any function such that 

    lim sup_ξ→∞ξ^-α/f(ξ) = 0   for all α > 0,

then there exists a measure μ_f supported on the set of Liouville numbers such that |μ̂_f(ξ)| ≲ f(|ξ|) for all ξ; on the other hand, if g : ℝ^+ →ℝ^+ is any function such that 

    lim inf_ξ→∞ξ^-α/f(ξ) > 0   for some α > 0,

then there does not exist a measure μ_g supported on the set of Liouville numbers such that |μ̂_g(ξ)| ≲ g(|ξ|) for all ξ∈ℝ.


§ CONVOLUTION STABILITY LEMMAS
 
The proofs of the main results of this paper rely on the construction of a sequence of functions which will approximate the measures that satisfy the statements of the theorem. The functions of the sequence are themselves a product of functions. In the frequency space, these products become convolutions and a major component of the proof is show that the Fourier decay estimates of these functions remain stable as the number of convolutions tends to infinity. The following two lemmas will be referred to when making an argument for stability by induction. This first lemma will be applied to Theorem <ref> and Theorem <ref>.
(Convolution Stability Lemma)
Let ω : ℕ→ℝ^+ be a function that increases to infinity such that ω(t) ≤log t for t ≥ 2. Suppose that N_2 is sufficiently large depending on ω and N_1. Moreover, let G,H : ℤ→ℂ be functions satisfying the following bounds for some N_3 > 1/ψ (β(N_2))^2:

 rCll
|G(s)|     ≤    1       for all s ∈ℤ 

G(0)     =    1    

G(s)     =     0       0 < |s| ≤N_2 

|G(s)|     ≲    θ(|s|)       everywhere 
 
|G(s)|     ≲    exp(-1/2|s/2N_3|^σ+1/4 σ)      when  |s|≥2N_3 

|H|     ≤    2     
 
|H(s)|    ≲     θ(|s|) ω(|s|)       everywhere 
 
|H(s)|     ≲    exp(-1/2|s/8N_1|^σ+1/4 σ)       when  |s|≥8N_1. 


Then

 rCll
|H*G(s) - H(s)|     ≲    N_2^-99       when  0≤|s| < N_2/4 
 
|H*G(s)|     ≲    θ(|s|) ω(|s|)       everywhere 
 
|H*G(s)|     ≲    exp(-1/2|s/8N_3|^σ+1/4 σ)       when  |s| ≥8N_3. 




A different version of this lemma will be applied to prove Theorem <ref>.
 
Let δ < 1/N_1^2. Suppose that N_2 is sufficiently large depending on ω and N_1. Moreover, let G,H : ℤ→ℂ be functions satisfying the following bounds for some N_3 > 1/ψ (β(N_2)):

 rCll
|G(s)|     ≤    1       for all s ∈ℤ 

G(0)     =    1    

G(s)     =     0       0 < |s| ≤N_2 

|G(s)|     ≲    δ      s ≠0 
 
|G(s)|     ≲    exp(-1/2|s/2N_3|^3/4)      when  |s|≥2N_3 

|H|     ≤    2     
 
|H(s)|     ≲    exp(-1/2|s/8N_1|^3/4)       when  |s|≥8N_1. 


Then

 rCll
|H*G(s) - H(s)|     ≲    N_2^-99       when  0 ≤|s| < N_2/4 
 
|H*G(s)|     ≲    δ^1/2       when |s| ≥ N_2/4 
 
|H*G(s)|     ≲    exp(-1/2|s/8N_3|^3/4)       when  |s| ≥8N_3. 


Before proving these lemmas, we need a preliminary estimate on θ. We will show that the function θ(ξ) decays like ξ^-1/τ up to an ϵ-loss in the exponent.

Let ψ, χ be as in Theorem <ref>, and let θ(ξ) be as in (<ref>). Then θ(|ξ|) ≈ |ξ|^-1/τ for large |ξ|. 


Since ψ(q) ≈ q^-τ by assumption, we have that ψ^-1(t) ≈ t^-1/τ. A similar argument shows that χ(t) ≈ 1. Hence χ(ψ^-1(1/|ξ|)) ≈ 1. Thus

    1/ψ^-1(1/|ξ|) χ(ψ^-1(1/|ξ|))≈ |ξ|^-1/τ.




 §.§ Proof of Lemma <ref>

 First, we prove (<ref>). Assume that 0 ≤ |s| ≤ N_2/4. Rewrite the expression as
 rCl
|H * G (s) - H (s)|     =     |∑_t∈ℤ H(s-t)G (t) - H (s)|

   =   |H(s)G(0) -H(s) +∑_t ≠0 H(s-t)G (t)| 

   ≤    ∑_t ≠0 |H(s-t)G (t)|.


Observe that we need only consider summands such that |t| ≥ N_2 because G(t) = 0 for |t| < N_2. The previous expression becomes

    ∑_|t| ≥ N_2 |H (s-t) G (t)|.


Apply the bound (<ref>) to |G(t)|. Notice that |s-t| ≥ |t|/2 ≫ N_1 when |s| < N_2/4. We may apply (<ref>) with |t|/2 in place of s to get an upper bound given that the bounding function is decreasing. Hence
 rCl
∑_|t| ≥N_2 |H (s-t) G (t)|    ≲    ∑_|t| ≥N_2exp(-1/2|t/8N_1|^σ+1/4 σ)

    ≤    N_2^-99.


The last inequality holds provided that N_2 is sufficiently large depending on N_1. The next task is to prove the estimate (<ref>). For 0 < |s| < N_2/4, the estimate follows from (<ref>) (<ref>). Indeed, the difference

    |H*G(s) - H(s)| ≲ N_2^-99≲ |s|^-99.

By the estimate (<ref>), we have that

    |s|^-99≲θ(|s|).


Now assume that |s| ≥ N_2/4. We have the inequality

    |H * G (s)| ≤I + II,

where

    I = ∑_|t| < 2N_1 |H (t)G (s-t)|,

and

    II = ∑_|t| ≥ 2N_1 |H (t)G (s-t)|.

Beginning with the sum I, we apply (<ref>) and observe that |s-t| ≥ |s| -|t| ≥ |s|/2 when |s| ≥ N_2/4. Then we may apply (<ref>) with |s|/2 in place of |s| to get
 rCl
I    ≲    θ(|s|) ∑_|t| < 2N_1 1

    ≲    θ(|s|) ω(|s|),

provided that N_2 is sufficiently large depending on ω and N_1 so that the final inequality holds. To bound the sum II, write


    II = A + B,

where 

    A = ∑_|t| ≥ 2N_1
    |s-t| ≤ |s|/2 |H (t)G (s-t)|,

and 

    B = ∑_|t| ≥ 2N_1
    |s-t| > |s|/2 |H (t)G (s-t)|.


To estimate the sum A, we apply (<ref>) and (<ref>). Observe that |s-t| ≤ |s|/2 implies that |t| ≥ |s|/2. Thus, t ≫ N_1 when |s| ≥ N_2/4. Therefore

    A ≲∑_|t| ≥ |s|/2exp(-1/2|t/8N_1|^σ+1/4 σ).

By the integral test, we get the following upper bound for A:

    A ≲∫_|s|/2^∞exp(-1/2|t/8N_1|^σ+1/4 σ)  dt.

Observe that the integrand is decaying nearly exponentially. From (<ref>), we may conclude

    A ≲θ(|s|).


For the sum B, we apply (<ref>) to G. Additionally, we may apply (<ref>). Since |s|/2 ≫ N_1, we have after making the substitution u = s - t that 
 rCl
B     ≲     θ(|s|)  ∑_|u| ≥|s|/2 exp(-1/2|u/8N_1|^σ+1/4 σ)

    ≲    θ(|s|)

where the last inequality is implied by

    ∑_|u| ≥ |s|/2exp(-1/2|u/8N_1|^σ+1/4 σ) ≤ 1.


Combining the bounds for I and II completes the proof for (<ref>). We turn now to proving (<ref>). Assume |s| ≥ 8N_3. We decompose the convolution as

    |H * G (s)| ≤I + II,

where 

    I = ∑_|t| < |s|/2 |H (s-t)G (t)|

and

    II = ∑_|t| ≥ |s|/2 |H (s-t)G (t)|.


Starting with I, we apply (<ref>). Then apply (<ref>) with s-t in place of s, and use the fact that |s-t|≥ |s|/2. Then

    I≲∑_|t| < |s|/2exp(-1/2|s/16N_1|^σ+1/4 σ).

There are at most |s|/2 summands in the above sum. Therefore

    I≲ |s| exp(-1/2|s/16N_1|^σ+1/4 σ).

We may bound the prior estimate by a single exponential function by choosing a smaller negative power and eliminating the linear term. Hence,

    I≲exp(-1/2|s/8N_3|^σ+1/4 σ).


For the sum II, apply the bounds (<ref>) and (<ref>) to get
 rCl
II     ≲    ∑_|t| > |s|/2 exp(-1/2|t/2N_3|^σ+1/4 σ).

To bound the above sum, we use the integral test. Thus

    II≲∫_t > |s|/2exp(-1/2| t/2 N_3|^σ + 1/4 σ)  dt.

To estimate this integral, we begin with a substitution. Let 

    u = 1/2|t/2N_3|^σ+1/4 σ.

Then

    du = σ +1/16 σ N_3|t/2N_3|^-3σ+1/4 σ dt.

The integral may be rewritten as

    16 σ N_3/σ +1∫_t = |s|/2^∞exp(-u) (2u)^3σ -1/σ +1 du.

Integrating by parts yields

    (16 σ N_3/σ +1) ( -exp(-u)(2u)^3σ -1/σ +1 |_t = |s|/2^∞ + 6σ - 2/σ + 1∫_t = |s|/2^∞exp(-u) (2u)^3σ -1/σ +1-1 du).


It is easy to see that expression above is dominated by the first term and the integral is an error term. Indeed, repeated integration by parts will yield relatively small terms (we are only interesting in finding an estimate up to a multiplicative constant) which contribute a negligible amount to the estimate. We consider only the first term in the estimate and evaluating it at the endpoints to get


    II≲ N_3 exp(-1/2|s/4N_3|^σ+1/4 σ)|s/4N_3|^3σ-1/4 σ.


Observe that exponential term is dominant for large values of s. We may bound the above expression by a single exponential term by choosing a smaller negative power. Hence,

 rCl
II     ≲    N_3 exp(-1/2|s/4N_3|^σ+1/4 σ)|s/4N_3|^3σ-1/4 σ

    ≤    exp(-1/2|s/8N_3|^σ+1/4 σ).

Combining the estimates on I and II completes the proof of the lemma.



 §.§ Proof of Lemma <ref>


The proof of Lemma <ref> shares many similarities with the proof of Lemma <ref>.

Beginning with (<ref>), assume |s| ≥ N_2/4 and write

    |H * G (s) - H (s)| ≤∑_|t| ≥ N_2 |H (s-t) G (t)|.

Apply the estimate (<ref>) with |t|/2 in place of s and the estimate (<ref>). Then
 rCl
|H * G (s) - H (s)|     ≲    δ∑_|t| ≥N_2exp(-1/2|t/8N_1|^3/4)
 
    ≤    δ
since the sum may be bounded above by 1.

In order to prove the estimate (<ref>), we assume |s| ≥ N_2/4. Write

    |H * G (s)| ≤I + II,

where 

    I = ∑_|t| < N_1 |H (t)G (s-t)|

and

    II = ∑_|t| ≥ N_1 |H (t)G (s-t)|.


For the sum I, apply the estimates (<ref>) and (<ref>). Then
 rCl
I     ≲    δ∑_|t| < N_1 1

    ≲    δN_1

    <     δ^1/2.

where the final inequality follows from the fact that δ < 1/N_1^2.

For the sum II, consider the term where s=t separately from other summands. Write

    II = ∑_|t| ≥ N_1
    s≠ t |H (t)G (s-t)| + |H(s)G(0)|.

Apply the estimates (<ref>), (<ref>) and (<ref>). Then
 rCl
II    ≲    δ∑_|t| ≥N_1
 s≠t exp(-1/2|t/8N_1|^3/4)  + exp(-1/2|s/8N_1|^3/4)

    ≲    δ^1/2.

The last inequality is implied by the bounds

    ∑_|t| ≥ N_1
    s≠ texp(-1/2|t/8N_1|^3/4) ≲ 1

and

    exp(-1/2|s/8N_1|^3/4) ≲δ^1/2.


For the final estimate (<ref>), assume |s|≥ 8N_3 and write

    |H * G (s)| ≤I + II,

where 

    I = ∑_|t| < |s|/2 |H (s-t)G (t)|

and

    II = ∑_|t| ≥ |s|/2 |H (s-t)G (t)|.


For the sum I, use the fact that |s - t| ≥ |s|/2 and apply (<ref>) and (<ref>) with |s|/2. Then
 rCl
 I     ≲    ∑_|t| < |s|/2 exp(-1/2|s/16N_1|^3/4)
 
     ≲    exp(-1/2|s/8N_3|^3/4).


For the sum II, apply (<ref>) and (<ref>) with |t| in place of s. Then
 rCl
 II     ≲    ∑_|t| > |s|/2 exp(-1/2|t/2N_3|^3/4)
 
     ≤    exp(-1/2|s/8N_3|^3/4).

This completes the proof of Lemma <ref>.




§ DOUBLING FUNCTIONS


If f : ℝ^+ →ℝ^+ is a decreasing or eventually decreasing function, we say that f is doubling if f(ξ/2) ≲ f(ξ) for all sufficiently large ξ.

We will need a few basic facts about doubling functions.

The function θ(ξ) is doubling.


The fact that θ(ξ) ≈ξ^-1/τ implies that θ(ξ) is eventually decreasing. To see that θ(ξ) is doubling, note that for sufficiently large q_1 and q_2 with q_1 < q_2, we have the assumption (<ref>), which is reproduced below for convenience.

    ψ(q_1)/ψ(q_2)≥(q_2/q_1)^σ.

Since ψ^-1 is decreasing, we have that ψ^-1(1/ξ) > ψ^-1(2/ξ). If ξ is sufficiently large that (<ref>) applies with q_1 = ψ^-1(2/ξ) and q_2 = ψ^-1(1/ξ), then we have

    ψ(q_1)/ψ(q_2) = 2/ξ/1/ξ≥(ψ^-1(1/ξ)/ψ^-1(2/ξ))^σ.

Hence
Cl
    θ(ξ/2)/θ(ξ) 

=     ψ^-1(1/ξ) χ(ψ^-1(1/ξ))/ψ^-1(2/ξ) χ(ψ^-1(2/ξ)) 

≤    (ψ^-1(1/ξ)/ψ^-1(2/ξ) )^1 + ϵ 

≤    2^(1 + ϵ)/σ.

Hence θ(ξ) is doubling.

Next, we show under very general conditions that a function with limit 0 must admit a decreasing, doubling majorant.

Suppose that M: ℤ→ℂ is any function such that |M(s)| → 0 as |s| →∞. Then there is a decreasing function N: ℝ^+ →ℝ^+ such that N(ξ) → 0 as ξ→∞ satisfying the doubling property such that |M(s)| ≤ N(|s|) for all s ∈ℤ.


First, we replace M by a decreasing function M_1 : ℝ^+ →ℝ^+ as follows. For s ∈ℕ, define

    M_1(s) = sup_|t| ≥ s |M(t)|.

Then M_1 is decreasing on [0,∞), |M_1(s)| ≤ M(|s|) for all s ∈ℤ, and lim_s →∞ M_1(s) = 0.

We construct N by taking the average of M. For ξ∈ℝ^+, define

    N(ξ) = 1/⌊ξ⌋ + 1∑_t ∈ℕ
     t ≤ξ M_1(t).

As N is an average of a decreasing function, it follows that N is decreasing; moreover, since M_1(t) → 0 as t →∞, it follows that N(ξ) → 0 as ξ→∞. Furthermore, it is easy to see that M_1(s) ≤ N(s) for s ∈ℕ:
rCl
N(s)     =     1/s + 1 ∑_t=0^s M_1(t) 

    ≥    1/s + 1 ∑_t = 0^s M_1(s) 

    =     1/s + 1 (s + 1) M_1(s) 

    =     M_1(s),

So |M(s)| ≤ M_1(|s|) ≤ N(|s|)) for all s ∈ℤ. 

It only remains to verify that N(s) has the doubling property (<ref>). We have for s ≠ 0 that
rCl
N(s/2)     =     1/⌊s/2 ⌋+ 1 ∑_t ≤s/2 
 t ∈ℕ M_1(t) 

    ≤      1/⌊s/2 ⌋+ 1 ∑_t ≤⌊s/2 ⌋
 t ∈ℕ M_1(t) + 1/⌊s/2 ⌋+ 1 ∑_⌊s/2 ⌋+ 1 ≤t ≤2 ⌊s/2 ⌋+ 1 
 t ∈ℕ M_1(t)  

    ≤      1/⌊s/2 ⌋+ 1 ∑_t ≤⌊s/2 ⌋
 t ∈ℕ M_1(t) + 1/⌊s/2 ⌋+ 1 ∑_⌊s/2 ⌋+ 1 ≤t ≤s + 1 
 t ∈ℕ M_1(t)  

    ≤    2/2 ⌊s/2 ⌋+ 2 ∑_t ≤s + 1 
 t ∈ℕ M_1(t). 

    ≤   2/s ∑_t ≤s + 1 
 t ∈ℕ M_1(t) 

    ≤    2/s ∑_t ≤s 
 t ∈ℕ M_1(t) + 2/s M_1(s + 1) 

    ≤    2/s ∑_t ≤s 
 t ∈ℕ M_1(t) + 2/s ∑_t ≤s 
 t ∈ℕ  M_1(t) 

    ≤    4/s ∑_t ≤s 
 t ∈ℕ M_1(t) 

    ≤    8/s + 1 ∑_t ≤s 
 t ∈ℕ M_1(t) 

    =     8 N(s),

as desired.




§ SINGLE-FACTOR ESTIMATES



 §.§ Single-factor estimates for Theorem <ref> and Theorem <ref>

In this section, we construct a function g_k with its support contained in intervals centered at rational numbers with denominator close to some number M_k. Let ψ(q) be a function satisfying (<ref>) and (<ref>). Suppose χ(q) is a function satisfying (<ref>). In the case of Theorem <ref>, we take χ(q) ≡ 1. 

Let M_k be a large positive integer. We choose an integer β(M_k) and a positive real number C_k so that


    1 ≤∑_M_k ≤ q ≤β(M_k) 
     q  prime1/q χ(q) = C_k ≤ 2.


The support of g_k will be contained in a family of intervals centered at rational numbers whose denominator is a prime number between M_k and β(M_k). 

We choose a nonnegative function ϕ∈ C_c^∞ with support in the interval [-1/2,0] satisfying the conditions

    ϕ̂(0) = 1

and

    ϕ̂(s) ≲exp( -|s|^σ + 1/2 σ).

The existence of such a function is guaranteed by a result of Ingham <cit.>.

Let 

    ϕ_p,q(x) = 1/q^2χ(q) ψ (q)ϕ(1/ψ(q)(x-p/q))

Now define

    g_k (x) = C_k^-1∑_M_k ≤ q < β(M_k) 
     q  prime∑_p = 1^qϕ_p,q(x).

Observe that the function g_k is supported on the interval [0,1].
 
Suppose g_k is defined as above. Then we have the following estimates for s ∈ℤ.
rCll 
ĝ_k(0)     =     1  

ĝ_k(s)     =     0       if 0 < |s| < M_k  

|ĝ_k(s)|     ≲    θ(|s|)       if s ≠ 0  

|ĝ_k(s)|     ≲    exp(-1/2 (ψ(β(M_k))^2|s|)^σ+ 1/4 σ )       if |s| ≥ψ(β(M_k))^-2.



A simple calculation gives us that


    ĝ_k (s) = C_k^-1∑_M_k ≤ q < β(M_k) 
     q  prime1/q^2 χ(q)∑_p = 1^q(p/qs) ϕ̂(ψ(q) s)

where (u) = ^-2 π i u. The sum in p has the value

    ∑_p=1^q (p/q s ) = 
        q    if q | s
    
        0    if q ∤ s.


Therefore, if s = 0, then the above sum will be equal to 1, and if 0 < |s| < M_k, then the above sum will vanish. This proves (<ref>) and (<ref>). Thus,


    ĝ_k (s) = C_k^-1∑_M_k ≤ q < β(M_k) 
     q  prime
     q | sϕ̂(ψ(q) s)/qχ(q).


For |s| ≥ M_k, we split the above sum into three pieces according to the size of q. We write


    ĝ_k (s) = C_k^-1 (I + II + III),

where
rCl
I     =     ∑_q ≥ψ^-1(1/|s|) 
 q  prime 
 q |s ϕ̂(ψ(q) s)/qχ(q), 

II     =      ∑_ψ^-1 (1/ √(|s|)) ≤q ≤ψ^-1(1/|s|) 
 q  prime 
 q |s ϕ̂(ψ(q) s)/qχ(q), 

III     =      ∑_q < ψ^-1(1/√(|s|)) 
 q  prime 
 q |s ϕ̂(ψ(q) s)/qχ(q).




  
*Estimate for I. For the sum I, we observe that the number of summands is ≲ 1. This observation is a consequence of assumption (<ref>) since it is implied that for a large enough q depending on ϵ we have,

    q^-τ-ϵ≤ψ(q) ≤ q^-τ + ϵ

which gives us

    t^-1/τ + ϵ≤ψ^-1(t) ≤ t^- 1/τ - ϵ

since ψ is decreasing. Taking logarithms, we conclude

    1/τ + ϵlog |s| ≤logψ^-1(1/|s|) ≤1/τ - ϵlog |s|.

Hence, the number of summands in the sum I is at most log |s|/logψ^-1(1/|s|)≲ 1. 

Apply the bound ϕ̂(ψ(q) s) ≤ 1 to each summand to get

    ∑_q ≥ψ^-1(1/|s|) 
     q  prime
     q | sϕ̂(ψ(q) s)/qχ(q)≲θ(|s|).




  
*Estimate for II. For the sum II, we observe that there are ≲ 1 summands by a similar argument as for the sum I. We apply the bound (<ref>) to show that the summand is bounded above by

    exp(- |ψ(q) s|^σ + 1/2 σ)/q χ(q)

If q = ψ^-1(1/|s|), then  

    exp(- |ψ(q) s|^σ + 1/2 σ)/q χ(q)≲θ(|s|).


It is enough to show for each q < ψ^-1(1/|s|) that 


    exp (- | ψ(q + 1) s|^σ + 1/2 σ)/(q + 1) χ(q + 1) - exp(- |ψ(q) s|^σ + 1/2 σ)/q χ(q) > 0.

If the inequality (<ref>) holds for all q < 1/|s|, then the summand is increasing in this domain, and is therefore maximized when q = 1/|s|, establishing the bound (<ref>) for such q.

In order to establish (<ref>), it is enough to verify that the numerator of the difference is positive. This numerator is

    exp (- |ψ(q + 1) s|^σ + 1/2 σ) q χ(q) - exp( - |ψ(q) s|^σ + 1/2 σ) (q + 1) χ(q + 1).

Since the logarithm is an increasing function, it is enough to show that 

    -|ψ(q + 1) s|^σ + 1/2 σ + log q + logχ(q) > - |ψ(q) s|^σ + 1/2 σ + log (q + 1) + logχ(q + 1).

This inequality is equivalent to

    log (q + 1) - log q + logχ(q + 1) - logχ(q) < |s|^σ + 1/2 σ (ψ(q)^σ + 1/2 σ - ψ(q + 1)^σ + 1/2 σ).

The Taylor series for the logarithm guarantees that log (q + 1) - log q = 1/q + O (1/q^2); the subpolynomial growth condition (<ref>) guarantees that logχ(q + 1) - logχ(q) = o (1/q). In total, the left side of inequality (<ref>) is 1/q + o (1/q). On the other hand, since we are in the regime where q < ψ^-1(1/|s|), the right side of (<ref>) is bounded below by

    |s|^σ + 1/2 σ (ψ(q)^σ + 1/2 σ - ψ(q + 1)^σ + 1/2 σ) ≥ 1 - (ψ(q + 1)/ψ(q))^σ + 1/2 σ.

By (<ref>), we have

    (ψ(q + 1)/ψ(q))^σ + 1/2 σ≤(q/q + 1)^σ + 1/2.

Hence,
 rCl
1 - (ψ(q + 1)/ψ(q))^σ+ 1/2 σ     ≥    1 - (q/q + 1 )^σ+ 1/2 

    =     (σ+ 1/2) 1/q + o (1/q^2 ).  


Since σ + 1/2 > 1, we see that the inequality (<ref>) holds for ψ(1 / √(|s|)) ≤ q ≤ψ(1 / |s|) provided that M_k (and hence |s|) is sufficiently large.

Hence we have the estimate

    II≲θ(|s|).



  
*Estimate for III. For the final sum, we apply the estimate (<ref>) to ϕ̂ to get
 rCl
∑_q <ψ^-1(1/√(|s|)) 
 q  prime 
 q |s ϕ̂(ψ(q) s)/qχ(q)    ≲    ∑_q < ψ^-1(1/ √(s)) 
 q  prime 
 q |s exp(-| ψ(q) s |^σ+ 1/2 σ)/q  

    ≤    ∑_q <ψ^-1(1/√(|s|)) 
 q  prime 
 q |s exp(-|s |^σ+ 1/4 σ)/q

    ≤    exp(-|s |^σ+ 1/4 σ) log(ψ^-1(1/√(|s|)))

    ≲    θ(|s|).


For the estimate (<ref>), we observe that |s| is sufficiently large for the estimate (<ref>) to apply to ϕ̂ for every q ∈ [M_k, β(M_k)]. As such
 rCl
|ĝ_k(s)|     ≲   ∑_M_k ≤q ≤β(M_k) 
 q  prime 
 q |s exp(-(ψ(q)|s|)^σ+1/2σ)/χ(q)q

    ≤    ∑_M_k ≤q ≤β(M_k) 
 q  prime 
 q |s exp(-(ψ(β(M_k))|s|)^σ+1/2σ)/χ(M_k)M_k.


The inequality |s| ≥ψ(β(M_k))^-2 gives us ψ(β(M_k)) ≤1/√(|s|). Therefore,

    |ĝ_k(s)|  ≲∑_M_k ≤ q ≤β(M_k) 
     q  prime
     q | sexp(-|s|^σ +1/4σ)/χ(M_k)M_k.


Observe that the number of summands is less than β(M_k). Moreover, we may disregard the denominator, for large M_k, to derive an upper bound. Hence,

    |ĝ_k(s)| ≲β(M_k) exp(-|s|^σ +1/4σ).


Now, we need to eliminate the β(M_k) term from the estimate but this will be at the cost some decay from the exponent. Rewrite the above inequality as 
 rCl
|ĝ_k(s)|    ≲    β(M_k) exp(-1/2|s|^σ+1/4σ)exp(-1/2|s|^σ+1/4σ) 

    ≤     β(M_k) exp(-1/2ψ(β(M_k))^-σ+1/2σ)exp(-1/2|s|^σ+1/4σ) 

when we apply |s| ≥ψ(β(M_k))^-2. From the equation (<ref>), when M_k is large enough we have

    1/2τ≤ -logψ(β(M_k))/logβ(M_k)≤ 2 τ

which may be rewritten as

    -1/2τlogβ(M_k) ≥logψ(β(M_k)) ≥ -2 τlogβ(M_k).

Exponentiating gives

    β(M_k)^-1/2τ≥ψ(β(M_k)) ≥β(M_k)^-2 τ.


Applying the upper bound from  the equation (<ref>) to (<ref>), we get

    |ĝ_k(s)| ≲β(M_k) exp(-1/2β(M_k)^τ(σ +1)/4σ)exp(-1/2|s|^σ +1/4σ).

For large M_k, we observe that the exponential term dependent on M_k is decaying much faster than β(M_k). Hence,
rCl
|ĝ_k(s)|     ≲    exp(-1/2|s|^σ+1/4σ) 

    ≲    exp(-1/2 (ψ(β(M_k))^2 |s|)^σ+ 1/4 σ ).





 §.§ Single-factor estimate for Theorem <ref>

In the case of Theorem <ref>, it is more convenient to choose the function g_k to be supported in a neighborhood of rational numbers with different denominators at very different scales. Thus, only one denominator will meaningfully contribute to the value of |ĝ_k(s)|.

As in subsection <ref>, we begin by defining a smooth function ϕ with its support in the interval [-1/2, 0] satisfying the conditions 

    ϕ̂(0) = 1

and 

    ϕ̂(s) ≲exp(-|s|^3/4).


Let n_k be an increasing sequence of integers to be specified later. For a given k, we choose q_k,1, …, q_k,n_k of prime numbers as follows. First, we choose q_k,1 to be a large prime number. We choose the remaining q_k,j so that q_k,2≫1/Ψ(q_k,1), q_k,3≫1/Ψ(q_k,2),…, q_k,n_k≫1/Ψ(q_k,n_k-1). Furthermore, we also assume that for each j, we have 

    max(1/q_k,j, ψ(q_k,j) ) < 1/2ψ(q_k,j-1).
 
Define

    g_k(x) = 1/n_k∑_j = 1^n_k1/q_k,jψ(q_k,j)∑_p = 1^q_k,jϕ(1/ψ(q_k,j)(x-p/q_k,j)).

Then

    ĝ_k(s) = 1/n_k∑_j = 1^n_k1/q_k,j∑_p = 1^q_k,jϕ̂(ψ(q_k,j) s) (p/q_k,js).


Remove any terms for which q_k,j does not divide s to get

    ĝ_k(s) = 1/n_k∑_1 ≤ j ≤ n_k
     q_k,j| sϕ̂(ψ(q_k,j) s) .



Suppose that g_k is defined as above. Then we have the following estimates for s ∈ℤ.
rCll
ĝ_k(0)     =     1  

ĝ_k(s)     =     0       if 0 < |s| < q_k,1  

|ĝ_k(s)|     ≲    1/n_k       s ≠0 

|ĝ_k(s)|     ≲    exp(-1/2 |ψ(q_k,n_k) s|^3/4 )       if |s| ≥ψ(q_k,n_k)^-1.



First, it is clear from (<ref>) and (<ref>) that ĝ_k(0) = 1, establishing (<ref>). Moreover, the sum (<ref>) is seen to be empty if 0 < |s| < q_k,1, establishing (<ref>).

To prove (<ref>), we split the sum (<ref>) depending on the size of q_k,j relative to s. Suppose j_0(s) is such that ψ(q_k,j_0)|s| > 1, but such that ψ(q_k,j_0 + 1)|s| ≤ 1, taking j_0(s) = 0 if ψ(q_k,1 )|s| < 1 or j_0 = n_k if ψ(q_k, n_k)|s| > 1.

    |ĝ_k(s)|  ≤1/n_k∑_j_0(s) + 1 ≤ j ≤ n_k 
     q_k,j| s|ϕ̂(ψ(q_k,j) s)| + 1/n_k∑_1 ≤ j ≤ j_0(s) 
     q_k,j| s|ϕ̂(ψ(q_k,j) s)|



For the second sum, we may apply (<ref>), the Schwartz tail for ϕ̂. Hence, using the assumption (<ref>),
 rCl
1/n_k ∑_1 ≤j ≤j_0(s) 
 q_k,j |s |ϕ̂(ψ(q_k,j) s) |     ≲    1/n_k ∑_1 ≤j ≤j_0(s) exp(-| ψ(q_k,j)s|^3/4) 

    ≲    1/n_k ∑_1 ≤j ≤j_0(s) exp( -2^3(j_0(s) - j)/4 ) 

    ≲    1/n_k.


For the first sum, recall that j_0 is chosen so that ψ(q_k,j_0 + 1)s < 1. Since q_k, j≥1/ψ(q_k,j_0 + 1) for any j ≥ j_0 + 2, it follows that 1/q_k,j s < 1 for such j. This means that it is impossible for q_k,j to divide s for j > j_0 + 1. Hence, the only term that can contribute to the sum is the j = j_0 + 1 term. To control the contribution of this term, we simply apply the bound  

    |ϕ̂(ψ(q_k,j) s)| ≤ 1

to bound the second sum by a constant times 1/n_k. Thus, for any integer s ≠ 0, we have the bound

    |ĝ_k(s)| ≲1/n_k.

It remains to show the bound (<ref>). For s ≥ψ(q_k,n_k)^-1, we can in fact apply the Schwartz bound (<ref>) for ϕ to every summand in (<ref>). Hence 
rCl
|ĝ_k(s)|     ≤    1/n_k ∑_1 ≤j ≤n_k 
 q_k, j |s |ϕ̂(ψ(q_k,j) s)| 

    ≲    1/n_k ∑_1 ≤j ≤n_k exp( - | ψ(q_k,j) s|^3/4 ) 

    ≲    1/n_k ∑_1 ≤j ≤n_k exp(- |2^n_k - j ψ(q_k, n_k) s|^3/4 ) 

    ≲    exp(- |ψ(q_k, n_k) s|^3/4 ).




§ STABILITY AND CONVERGENCE OF Μ̂_Χ,Ω

In order to prove Theorems <ref>, <ref>, and <ref>, we will piece together the functions g_k provided in Section <ref> across multiple scales. Lemmas <ref> and <ref> are used to show that the Fourier transforms ĝ_k of the functions g_k do not exhibit much interference. The construction proceeds slightly differently in the case of Theorem <ref>, as this theorem does not prescribe a specific decay rate for μ̂. 


 §.§ Construction of μ for Theorem <ref> and Theorem <ref>

Let ψ and χ be functions satisfying the assumptions (<ref>), (<ref>), (<ref>), and (<ref>). Recall that in the case of Theorem <ref> that we take χ≡ 1, and we showed in Remark <ref> that ψ satisfies assumptions (<ref>) and (<ref>).
We begin by constructing a sequence of functions (μ_χ, ω, k)_k ∈ℕ where μ_χ, ω, k(x) is the product 


    μ_χ, ω, k(x) = ∏_i=1^k g_i(x).


For each g_i we choose an associated M_i such that the estimates in Lemma <ref> apply. We further assume that the M_i's are spaced sufficiently far apart to satisfy the conditions of Lemma <ref>. In particular, this implies that for each i ≥ 1 we have

    M_i+1≥ψ(β(M_i))^-2.


Taking the Fourier transform of this sequence, we get the sequence (μ̂_χ, ω, k)_k ∈ℕ where 

    μ̂_χ, ω, k (s) = ĝ_1 * ⋯ * ĝ_k (s).
 

With this sequence of functions defined, the next objective is to show that the sequence is uniformly convergent and that the functions μ̂_χ, ω, i satisfy a similar decay estimate (up to a constant) for all i. We begin with the latter:
 
For the sequence of functions (μ̂_χ, ω, k)_k ∈ℕ defined above, we have the following statements for any integers k, l with k > l:

 rCll
|μ̂_χ, ω, k(0)|     ≤      2       
 
|μ̂_χ, ω, l(s) -μ̂_χ, ω, k(s)|     ≲    ∑_j = l +1^k M_j^-99       when  0 ≤|s| < M_l/4

|μ̂_χ, ω, k(s)|     ≲      θ(|s|) ω(|s|)        for all s ≠ 0 

|μ̂_χ, ω, k(s)|     ≲       exp(-1/2 (ψ(β(M_k))^2|s|)^σ+ 1/4 σ )       if |s| ≥ψ(β(M_k))^-2. 

Note that since μ is a positive measure, (<ref>) implies that |μ̂_χ, ω, k(s)| ≤ 2 for all s.


We prove Lemma <ref> by induction and repeated application of Lemma <ref>. We begin with the basis by letting k = 2. Then μ̂_2 = ĝ_1 * ĝ_2. Apply Lemma <ref> with H = ĝ_1, G= ĝ_2, N_1 = ψ(β(M_1))^-2, N_2 = M_2 and N_3 = ψ(β(M_2))^-2. Then the estimates (<ref>), (<ref>) and  (<ref>) immediately follow from (<ref>), (<ref>) and (<ref>), respectively. The statement (<ref>) can be shown by the following calculation:
 rCl
|μ̂_χ, ω, 1(0)|     ≤    |ĝ_1(0) - ĝ_1* ĝ_2 (0)| + |ĝ_1(0)|

    ≤    𝒪(M_2^-99) + 1

    ≤    2

where the last inequality holds for the choice of a sufficiently large M_2. Now assume that Lemma <ref> holds for k up to n-1. Then for the case k = n, we make the choice H = μ̂_n-1 = ĝ_1*⋯*ĝ_n-1, G= ĝ_n, N_1 = ψ(β(M_n-1))^-2, N_2 = M_n and N_3 = ψ(β(M_n))^-2. From the induction hypothesis and Lemma <ref>, the estimates (<ref>) and (<ref>) immediately follow. For estimate (<ref>), assume l < k and assume |s| ≤ M_l/4. Then the triangle inequality gives:

    |μ̂_χ, ω, l(s) -μ̂_χ, ω, k(s)| ≤ |μ̂_χ, ω, l(s) - μ̂_χ, ω, l+1(s)| + ⋯ + |μ̂_χ, ω, k-1(s) - μ̂_χ, ω, k(s)|.


By the induction hypothesis, |μ̂_i(s) - μ̂_i+1(s)| ≲ M_i+1^-99 for l ≤ i ≤ k-2 and Lemma <ref> gives

    |μ̂_χ, ω, k-1(s) - μ̂_χ, ω, k(s)|  ≲ M_k^-99.
 
Consequently,

    |μ̂_χ, ω, l(s) -μ̂_χ, ω, k(s)| ≲∑_j = l+1^k M_j^-99.


Finally, from the calculation
 rCl
|μ̂_χ, ω, k(0)|     ≤    |μ̂_χ, ω, 1 (0) - μ̂_χ, ω, k (0)| + |μ̂_χ, ω, 1 (0)|

    ≤    1 +𝒪(∑_j = 2^k M_j^-99)

    ≤    2,

the estimate (<ref>) is proved. 

Turning now to proving the uniform convergence of the sequence (μ̂_χ, ω, k)_k ∈ℕ, we have the following lemma.

The sequence (μ̂_χ, ω, k)_k ∈ℕ converges uniformly for all s ∈ℤ to some function M(s). This function M(s) has the property that 

    |M(s)| ≲θ(|s|) ω(|s|); s ∈ℤ


 Let ϵ > 0. There exists an m_0, depending on ϵ and ω, sufficiently large such that 

    θ(|s|) ω(|s|) < ϵ/2C

when |s| ≥ M_m_0/4. Here C is taken to be the implicit constant for estimate (<ref>). Then for m≥ n≥ m_0 
 rCl
 |μ̂_χ, ω, m(s) - μ̂_χ, ω, n(s)|     ≤    |μ̂_χ, ω, m (s)| + |μ̂_χ, ω, n (s)|

     <     ϵ.

When 0 ≤ |s| ≤ M_m_0/4, applying the estimate (<ref>) gives

    |μ̂_χ, ω, m(s) - μ̂_χ, ω, n(s)|  ≤∑_j = n^∞ M_j^-99

and the sum may be made to be less than ϵ.

Hence the sequence μ̂_χ, ω, n has a uniform limit M(s). An upper bound on |M(s)| will follow from Lemma <ref>. Suppose |s| is such that M_k/4≤ |s| ≤M_k+1/4.

Then the estimate (<ref>) gives that

    |μ̂_χ, ω, k(s) | ≲θ(|s|) ω(|s|),

and (<ref>) and the triangle inequality give
rCl
|M(s)|     ≤    |μ̂_χ, ω, k(s)| + lim sup_l ≥k |μ̂_χ, ω, k(s) - μ̂_χ, ω, l(s)| 

    ≲    θ(|s|) ω(|s|) + ∑_j=k+1^∞ M_j^-99 

    ≲    θ(|s|) ω(|s|) + |s|^-99 

    ≲    θ(|s|) ω(|s|),

as desired.

In order to show that the sequence μ_χ, ω, n converges to a weak limit μ using the convergence of the μ̂_χ, ω, n(s), it is normal to appeal to a theorem such as the Lévy continuity theorem. However, this is slightly inconvenient as we only have estimates for μ̂_χ, ω, n(s) at integer values s. We will provide a proof of the weak convergence below. First, we will need the following technical lemma relating the Fourier series of a measure supported on the interval [0,1] to its Fourier transform. A stronger version of this lemma can be found as Lemma 1 of Chapter 17 in the book of  Kahane <cit.>.

Suppose that μ is a measure supported on the interval [0,1] satisfying an estimate of the form

    |μ̂(s)| ≲ N(|s|)   for all s ∈ℤ

where N: ℝ^+ →ℝ^+ is a non-increasing function satisfying the doubling property

    N(ξ/2) ≲ N(ξ)   for all ξ∈ℝ^+.

Then |μ̂(ξ)| ≲ N(|ξ|) for all ξ∈ℝ.

We have already seen that θ(ξ) ω(ξ) is a doubling function for ξ > 0. Thus we can apply Lemma <ref>.

The sequence of measures μ_χ, ω, k has a weak limit μ_χ, ω. This weak limit μ_χ, ω satisfies the estimate

    μ̂_χ, ω(ξ) ≲θ (|ξ|) ω(|ξ|)

for all real numbers ξ.


Observe that each measure μ_χ, ω, n has total variation norm bounded by 2. We claim that the measures μ_χ, ω, k have a weak limit. First, by the Banach-Alaoglu theorem, there exists a subsequence μ_χ, ω, n_k that has a weak limit μ_χ, ω.  Since each measure μ_χ, ω, n_k is supported in [0,1], the weak limit μ_χ, ω is supported in [0,1]. 

In particular, since μ is supported in [0,1], each Fourier coefficient μ̂(s) of μ is obtained by integrating against a continuous, compactly supported function. Hence, for each s ∈ℤ, lim_k →∞μ̂_χ, ω, n_k(s) = M(s), where M(s) is the limit in Lemma <ref>. 

By the corollary to Theorem 25.10 of Billingsley <cit.>, it is enough to check that each weakly convergent subsequence of {μ_χ, ω, n} converges weakly to μ_χ, ω. Suppose {ν_χ, ω, n} is a subsequence of the μ_χ, ω, n with some weak limit ν. Then ν is supported on [0,1], so by the same argument as in the previous paragraph, Lemma <ref> implies that ν̂(s) = M(s) for every s ∈ℤ. Since a measure supported on [0,1] is uniquely determined by its Fourier-Stieltjes series, it follows that ν = μ_χ, ω as desired.

Finally, we verify that μ̂_χ, ω(ξ) satisfies the estimate (<ref>). This estimate holds for integer values of s by the estimate (<ref>). Hence, Lemma <ref> shows that μ̂_χ, ω(ξ) satisfies the same estimate for ξ∈ℝ.

Hence the measures μ_χ, ω, k have a weak limit supported on [0,1]. We now verify that this weak limit is indeed supported on the set E(ψ).

Let μ be as in Lemma <ref>. Then μ is supported on E(ψ).


It is easy to see that

    ϕ_p,q⊂[p/q - 1/2ψ(q), p/q + 1/2ψ(q)]

and therefore

    g_k ⊂⋃_M_k≤ q ≤β(M_k) 
     q  prime⋃_p = 0^q-1[p/q - 1/2ψ(q), p/q + 1/2ψ(q)].

Since each μ_χ, ω, k is the product of g_i's its support is an intersection of these supports.

    μ_χ, ω, k⊂⋂_i = 1^k⋃_M_i≤ q ≤β(M_i) 
     q  prime⋃_p = 0^q-1[p/q - 1/2ψ(q), p/q + 1/2ψ(q)].

Because the measure μ_χ, ω is defined as the weak limit of the measures μ_χ, ω, k, we have the containment

    μ_χ, ω⊂⋂_i = 1^∞⋃_M_i≤ q ≤β(M_i) 
     q  prime⋃_p = 0^q-1[p/q - 1/2ψ(q), p/q + 1/2ψ(q)].

Observe that if x ∈μ_χ, ω and k ∈ℕ, then x must also lie in one of the intervals 

    [p/q - 1/2ψ(q), p/q + 1/2ψ(q)]

for some M_k ≤ q ≤β(M_k).

Therefore, there exists an infinite number of rational numbers p/q which satisfy

    | x-p/q| ≤ψ (q)

and we may conclude that μ⊂ E(ψ).

The measure μ_χ, ω satisfies all of the properties required to prove Theorem <ref>. Hence, the proof of Theorem <ref> is complete.

To show Theorem <ref>, it is also necessary to verify that the support of μ is contained in a set of generalized α-Hausdorff measure equal to zero. This will be shown in Section <ref>.


 §.§ Construction of μ for Theorem <ref>

We now construct the measure μ described in Theorem <ref>. The biggest difference between this construction and the one in the previous subsection is that we do not state explicit quantitative estimates describing the decay of the Fourier transform of the measures.

Choose a positive integer n_1 and let M_1 be a large integer. We will choose the sequences {n_j : j ≥ 2} and {M_j : j ≥ 2} to be rapidly increasing sequences of integers satisfying a certain set of conditions below. For each j, we choose prime numbers  q_j, 1, …, q_j,n_j with M_j ≤ q_j,1≪⋯≪ q_j, n_j. When we choose the M_j, we will impose the condition that M_j+1≫ q_j, n_j as well. Given q_j,1, …, q_j,n_j we define the function g_j as in Subsection <ref>.

We define the function μ_k to be the pointwise product

    μ_k(x) = ∏_j=1^k g_j(x)

so μ̂_1(s) = ĝ_1(s) and so that for any k ≥ 2

    μ̂_k(s) = ĝ_k(s) * μ̂_k-1(s).


We are now ready to state the main estimate on μ̂_k.

Suppose that the functions g_k are chosen as above. Then provided that the sequences n_j and M_j are chosen appropriately, the measures μ̂_k satisfy the following estimates for all integers k ≥ l. All implicit constants below are assumed to be independent of k and l.
rCll
|μ̂_k(0)|     ≤    2  

|μ̂_k(s) -  μ̂_l(s)|     ≤    ∑_j=l+1^k M_j^-99       if 0 ≤ |s| ≤ M_l/4  

|μ̂_k(s)|     ≲     n_k^-1/2       if |s| ≥ M_k/4 

|μ̂_k(s)|     ≲    exp(-1/2 |1/8 ψ(q_k, n_k) s |^3/4 )       if |s| ≥ 8ψ(q_k,n_k)^-1.



Let n_1 and M_1 be positive integers, and choose prime numbers q_1, 1, …, q_1, n_1 such that 1 ≤ q_1,1 < q_1,2 < ⋯ < q_1, n_1 satisfy the conditions of Lemma <ref>. Then ĝ_1 satisfies the estimates of Lemma <ref> and in particular satisfies the estimates of Lemma <ref>.

Given g_1, …, g_k such that μ_k satisfies the four conditions above, we will describe how to choose the integers n_k+1 and M_k+1 and how to choose the function g_k+1 so that μ_k+1 will satisfy the four conditions above. Let N_1 = ψ(β(M_k))^-1. Lemma <ref> requires that the quantity δ is chosen so that δ < 1/N_1^2; hence, we select n_k+1 = 100 N_1^2. Choose M_k+1 = N_2 ≫ n_k to be a prime number that is sufficiently large to satisfy the conditions of Lemma <ref>. Take N_2 = q_k+1, 1 < ⋯ < q_k+1, n_k+1 sufficiently well-spaced to satisfy the conditions of Lemma <ref>. Then, choose N_3 = 1/ψ(q_k+1, n_k+1)≫ q_k+1,1. With these choices, we define g_k+1 as in Subsection <ref>. Hence Lemma <ref> implies that ĝ_k+1 satisfies the estimates required to serve as the function G in Lemma <ref>. 

Hence, we can apply Lemma <ref> with H = μ̂_k, G = ĝ_k+1, N_1 = ψ(q_k, n_k)^-1, N_2 = q_k+1,1, and N_3 = ψ(q_k+1, n_k+1)^-1, and δ = 1/n_k+1.

This implies the estimates
rCll
| μ̂_k+1(s) - μ̂_k(s)|     ≤     M_k+1^-99       if 0 ≤ |s| ≤M_k+1/4 

|μ̂_k+1(s)|     ≲    n_k+1^-1/2       if |s| ≥M_k+1/4 

|μ̂_k+1(s)|     ≲    exp(-1/2 |1/8 ψ(q_k+1, n_k+1) s |^3/4  )       if |s| ≥ 8ψ(q_k+1,n_k+1)^-1.

Hence μ̂_k+1 satisfies the estimates (<ref>) and (<ref>). In order to check (<ref>), assume l < k+1 and |s| ≤M_l/4. If l = k, then the inequality follows from (<ref>). If l < k, then applying the inductive assumption (<ref>) to estimate the difference μ̂_k - μ̂_l and applying (<ref>) to estimate the difference μ̂_k+1- μ̂_k gives
rCl
|μ̂_k+1(s) - μ̂_l(s)|     ≤    |μ̂_k+1(s) - μ̂_k(s)| + |μ̂_k(s) - μ̂_l(s)| 

    ≤    M_k + 1^-99 + ∑_j=l+1^k M_k^-99 

    =     ∑_j=l+1^k+1 M_k^-99.

This establishes (<ref>) for μ̂_k+1. Applying (<ref>) with l = 1 and s = 0, we see that
rCl
|μ̂_k+1(0)|     ≤    |μ̂_k+1(0) - μ̂_1(0)| +  |μ̂_1(0)| 

    ≤    ∑_j=2^k+1 M_j^-99 + 1 

    ≤    2

assuming the M_j grow sufficiently rapidly.


The sequence μ̂_k converges uniformly for all s ∈ℤ to a function M(s). This function M(s) has the property that |M(s)| → 0 as |s| →∞.


The proof is similar to that of Lemma <ref>. Let ϵ > 0. Because n_k →∞, there is an index k_0 such that n_k_0^-1/2 + ∑_j=k_0 + 1^∞ M_j^-99 < ϵ / 2C, where C is the implicit constant from Lemma <ref>. 

Suppose |s| > M_k_0/4, and choose l ≥ k_0 such that M_l/4≤ |s| < M_l + 1/4.  We have that |μ̂_l(s)| ≲ n_l^-1/2≤ n_k_0^-1/2 < ϵ/2. Hence for k ≥ l, we have
rCl
|μ̂_k(s)|     ≤    |μ̂_l(s)| + |μ̂_k(s) - μ̂_l(s)| 

    ≤    ϵ/2 + ∑_j=l+1^k M_j^-99 

    ≤    ϵ/2 + ∑_j=k_0 + 1^∞ M_j^-99 

    ≤    ϵ.

Hence |μ̂_k(s)| ≤ϵ for all |s| ≥M_k_0/4 and all k ≥ k_0. 

If |s| ≤M_k_0/4 and k_0 ≤ l ≤ k, then we have

    |μ̂_k(s) - μ̂_l(s)| ≤∑_j=l+1^k M_j^-99≤∑_j=k_0 + 1^∞ M_j^-99 < ϵ/2.
 
This proves that the sequence μ̂_k(s) is uniformly Cauchy and hence uniformly convergent. Let M(s) denote the uniform limit of this sequence.

Finally, we verify that M(s) → 0 as |s| →∞. Suppose |s| is such that M_k/4≤ |s| ≤M_k+1/4. Then we have from Lemma <ref> that |μ̂_k(s)| ≲ n_k^-1/2, and
rCl
M(s)     ≲    |μ̂_k(s)| + lim sup_l →∞ |μ̂_l(s) - μ̂_k(s)| 

    ≲    n_k^-1/2 + ∑_j=k+1^∞ M_j^-99 

    ≲    n_k^-1/2

since M_k+1≫ n_k.  Since the sequence n_k →∞, this shows that M(s) → 0 as |s| →∞, as desired.


The rest of this proof is similar to the proof of Theorem <ref>. In order to apply Lemma <ref>, we use the fact from Lemma <ref> that M is majorized by N(|s|), where N is a doubling function. This will allow us to apply Lemma <ref>.

We are now ready to show that the measures μ_k converge to a weak limit.


The sequence of measures μ_k has a weak limit μ. This weak limit μ satisfies the estimate

    |μ̂(ξ)| → 0   as |ξ| →∞ in ℝ.

Hence μ is a Rajchman measure.


This proof is almost exactly the same as the proof of Lemma <ref>, but when we apply Lemma <ref>, we use N(|s|) as the bound on M(s), where N(s) is the function constructed in Lemma <ref>.



Let μ be the weak limit in Lemma <ref>. Then the support of μ is contained in E(ψ).


This lemma can be shown in a similar manner to Lemma <ref>; we see that if x ∈(μ) then for each k, there exists a denominator q_k,j_k and a numerator p_k, j_k such that |x - p_k, j_k/q_k, j_k| ≤ψ(q_k, j_k); hence, x is ψ-well-approximable.

Therefore, the measure μ satisfies all of the properties promised in the statement of Theorem <ref>. Thus the proof of Theorem <ref> is complete.


§ A BOUND ON THE GENERALIZED HAUSDORFF MEASURE
 
To complete the proof of Theorem <ref>, we must show that F_α, which is taken to be the support of μ_k,ω, has zero α-Hausdorff measure.

Let F_α be a closed subset of

    {x : | x - r/q| < ψ(q)  for some integers 0 ≤ r ≤ q-1, M_k ≤ q ≤β(M_k), q  prime, 
      k ∈ℕ}.

Let ϵ > 0. Then there exists a cover 𝒰 of F_α by open intervals U such that

    ∑_U ∈𝒰α(U) < ϵ.
 


The set F_α satisfies the following containment:

    F_α⊂⋂_k=1^∞⋃_M_k ≤ q ≤β(M_k)
     q  prime⋃_p=0^q-1[p/q-ψ(q),  p/q+ψ(q)].

For any k the following collection of closed intervals is a cover for F_α:

    {[p/q-ψ(q),  p/q+ψ(q)]: M_k ≤ q ≤β(M_k), q  prime, 0 ≤ p ≤ q-1}.

Denote this collection as ℐ_k. The following collection 𝒰 is also a cover of F_α:

    𝒰 = {J⋂[p/q-ψ(q),  p/q+ψ(q)]: M_k ≤ q ≤β(M_k), q  prime, 0 ≤ p ≤ q-1; J ∈ℐ_k-1}.

Fix a prime number q with M_k ≤ q ≤β(M_k) and let J ∈ℐ_k-1. Observe that the intersection of J with the interval [p/q - ψ(q), p/q + ψ(q) ] is either empty or is a closed interval of length at most 2 ψ(q). We claim that the number of such intervals that intersect J satisfies

    #{p: [p/q-ψ(q),  p/q+ψ(q)]∩ J ≠∅}∼ |J|q,

where |J| denotes the length of the interval J.

The interval J belongs to ℐ_k-1. Therefore, |J| ≥ψ(β(M_k-1))^-2 by the assumption (<ref>). Hence |J| ≫1/M_k, and therefore, |J| ≫1/q.

The interval [p/q - ψ(q), p/q + ψ(q)] intersects J if and only if p/q lies in a ψ(q)-neighborhood of J. Since ψ(q) ≈ q^-τ by (<ref>) and τ > 2, we have that ψ(q) ≪ 1/q if k is sufficiently large. Hence, [p/q - ψ(q), p/q + ψ(q)] intersects J if and only if p/q lies in an interval J' of length |J'| = |J| + 2 ψ(q) ∼ |J|.

Write J' = [a,b]. Then the smallest multiple of p/q contained in J' is ⌈ q a ⌉/q, and the largest multiple of p/q contained in J' is ⌊ q b⌋/q. So the total number of multiples of p/q contained in J' is 
Cl
    ⌊qb ⌋- ⌈q a ⌉+ 1 

=     qb - qa + O(1) 

=     q|J'| + O(1) 

∼     q |J| + O(1).

Since |J| ≫ 1/q, we have q |J| ≫ 1. Therefore,

    #{p : [p/q - ψ(q), p/q + ψ(q) ] ∩ J ≠∅}∼ |J|q,

as claimed.

Then
 rCl
∑_U ∈𝒰 α((U))     ≤    ∑_J∈ℐ_k-1 ∑_M_k ≤q ≤β(M_k)
 q  prime ∑_0≤p ≤q-1 α((J⋂[p/q-ψ(q),  p/q+ψ(q)]))

    ∼    ∑_J∈ℐ_k-1 |J| ∑_M_k ≤q ≤β(M_k)
 q  prime q α(ψ(q)).    

From assumption (<ref>), α(ψ(q)) = q^-2. Therefore,

    ∑_J ∈ℐ_k-1 |J| ∑_M_k ≤ q ≤β(M_k)
     q  prime q α(ψ(q)) = ∑_J∈ℐ_k-1 |J| ∑_M_k ≤ q ≤β(M_k)
     q  prime q^-1.

By choosing β(M_k) = M_k^γ where γ > 1 is some positive number we get

    ∑_M_k ≤ q ≤β(M_k)
     q  prime q^-1∼loglogM_k^γ -  loglog M_k = logγ.

Consequently,
 rCl
∑_J∈ℐ_k-1 |J| ∑_M_k ≤q ≤β(M_k)
 q  prime q^-1    ∼    ∑_J ∈ℐ_k |J|

    ≲    ∑_M_k-1 ≤q ≤β(M_k-1)
 q  prime ∑_0 ≤p ≤q-1 ψ(q)

    =     ∑_M_k-1 ≤q ≤β(M_k-1)
 q  prime q ψ(q).

Recall that ψ(q) ≈ q^-τ, so
 rCl
∑_M_k-1 ≤q ≤β(M_k-1)
 q  prime q ψ(q)     ≈    ∑_M_k-1 ≤q ≤β(M_k-1)
 q  prime q^-τ+1

    ≲    M_k-1^-τ+2.

The exponent -τ + 2 < 0. Hence, if k is chosen to be sufficiently large, we have 

    ∑_U ∈𝒰α(U) < ϵ,

as desired.

myplain

