Fourier Dimension in Duffin--Schaeffer Conjecture

1. Problem Statement and Context

The Core Problem:
The paper addresses the determination of the Fourier dimension of a specific set of real numbers $W^*(\psi, \theta)$ defined by a generalized Duffin–Schaeffer type approximation condition.

Let $\psi: \mathbb{N} \to [0, 1/2)$ be a positive function and $\theta: \mathbb{N} \to \mathbb{R}$ be a real-valued function. Let $\{A_q\}$ and $\{B_q\}$ be integer sequences with $\gcd(A_q, B_q)=1$ and $B_q > 0$. The set $W^*(\psi, \theta)$ consists of all $x \in [0, 1]$ such that:
$$\left| x - \frac{p + \theta(q)}{q} \right| < \frac{\psi(q)}{q}$$
holds for infinitely many pairs $(p, q) \in \mathbb{Z} \times \mathbb{N}$ satisfying the coprimality constraint:
$$\gcd(B_q p + A_q, q) = 1.$$

Motivation and Gap:

• Metric Number Theory: While the Lebesgue measure and Hausdorff dimension of such sets are well-understood (e.g., the recent proof of the Duffin–Schaeffer conjecture by Koukoulopoulos and Maynard), the Fourier dimension remains a challenging frontier.
• Salem Sets: A set is called a Salem set if its Hausdorff dimension equals its Fourier dimension. Constructing deterministic Salem sets is difficult.
• Previous Limitations:
  • Kaufman and Bluhm established the Fourier dimension for the homogeneous case ($\theta=0$) with $\psi(q) = q^{-\tau}$ ($\tau \ge 1$), but their proofs relied heavily on the coprime restriction.
  • Chen and Xiong recently solved the Fourier dimension for the set $W(1, 1; \psi, 0)$ (without the coprime restriction) under specific divergence conditions, but the coprime version (the Duffin–Schaeffer setting) for general $\psi$ and inhomogeneous $\theta$ remained open.
  • Specifically, the Chen–Xiong conjecture regarding the validity of the dimension formula for irrational $\theta$ in the coprime setting was unresolved.

2. Methodology

The authors employ a combination of harmonic analysis, probabilistic methods, and number-theoretic estimates to establish both upper and lower bounds for the Fourier dimension.

A. Upper Bound (Proposition 2.1)

• Strategy: Proof by contradiction using the First Borel–Cantelli Lemma.
• Assumption: Assume the Fourier dimension is strictly greater than $2s(\psi)$, implying the existence of a measure $\nu$ with Fourier decay $|\hat{\nu}(\xi)| \ll |\xi|^{-s}$ for some $s > s(\psi)$.
• Technique:
  1. Estimate the measure $\nu(A_q)$ of the intervals satisfying the approximation condition.
  2. Use Parseval's identity to express $\nu(A_q)$ in terms of the Fourier coefficients of the indicator functions of these intervals and the Fourier transform of $\nu$.
  3. Key Number-Theoretic Tool: The authors derive a uniform upper bound for a generalized Ramanujan sum involving the coprimality condition $\gcd(B_q p + A_q, q) = 1$.
     • Lemma 2.3: They prove that $\left| \sum_{\substack{0 \le p \le q-1 \\ \gcd(B_q p + A_q, q)=1}} e(-k p/q) \right| \ll \gcd(q, k) \log q$.
  4. Show that under the assumption $s > s(\psi)$, the sum $\sum \nu(A_q)$ converges, implying $\nu(W^*) = 0$, which contradicts the assumption that $\nu$ is supported on $W^*$.

B. Lower Bound (Proposition 2.2)

• Strategy: Constructive approach using a Cantor-like measure (a variant of the Cai–Hambrook construction).
• Technique:
  1. Density Construction: Define a sequence of absolutely continuous measures with densities $g_M$. These densities are averages of periodic functions $\Phi^*_{q, \theta}$ constructed from a smooth bump function $f$.
  2. Recursive Selection: Select a sequence of integers $(M_k)$ recursively to ensure the stability of the Fourier decay.
  3. Fourier Decay Estimates:
     • Analyze the Fourier coefficients of the density functions $g_M$.
     • Use the Poisson Summation Formula to relate the spatial distribution to frequency decay.
     • Apply the Ramanujan sum bounds (Lemma 2.3) to control the off-diagonal terms in the Fourier expansion.
  4. Limit Measure: Define the limit measure $\nu = \lim \nu_k$. By ensuring the sequence of Fourier transforms is Cauchy in the supremum norm (using Lévy's continuity theorem), they establish that the limit measure $\nu$ has the desired Fourier decay rate: $|\hat{\nu}(\xi)| \ll (1+|\xi|)^{-(2s(\psi)-\epsilon)}$.
  5. Support: Verify that the support of the constructed measure lies within $W^*(\psi, \theta)$.

3. Key Definitions and Parameters

• Fourier Dimension ($\dim_F E$): Defined as the supremum of $s$ such that there exists a probability measure $\nu$ on $E$ with $|\hat{\nu}(\xi)| \ll (1+|\xi|)^{-s/2}$.
• Critical Exponent $s(\psi)$:
  $$s(\psi) = \inf \left\{ s \ge 0 : \sum_{q=1}^\infty \left( \frac{\psi(q)}{q} \right)^s < \infty \right\}$$
• Convergence Condition: The main result holds under the condition:
  $$\sum_{q \in \mathbb{N}} \psi(q) \prod_{\substack{p|q \\ p \nmid B_q}} (1 - p^{-1}) < \infty$$
  This ensures the Lebesgue measure of the set is zero, making the Fourier dimension the relevant metric.

4. Main Results

Theorem 1:
Let $\psi: \mathbb{N} \to [0, 1/2)$ be an arbitrary function satisfying the convergence condition (3). Then the Fourier dimension of the set $W^*(\psi, \theta)$ is:
$$\dim_F W^*(\psi, \theta) = \min\{2s(\psi), 1\}$$

Significance of the Result:

1. Generalization: It provides a complete characterization for the inhomogeneous case ($\theta \neq 0$) with coprime constraints ($\gcd(B_q p + A_q, q) = 1$).
2. Resolution of Conjecture: It affirmatively resolves the coprime formulation of the Chen–Xiong conjecture, confirming that the dimension formula holds even with the Duffin–Schaeffer type arithmetic restrictions.
3. Recovery of Classical Results: By setting $A_q=0, B_q=1$, the theorem recovers the classical results of Kaufman and Bluhm for the homogeneous case $\psi(q) = q^{-\tau}$ ($\tau \ge 1$).
4. Salem Sets: The result implies that for a wide class of approximation functions, these Diophantine sets are Salem sets (since the Fourier dimension matches the Hausdorff dimension, which is known to be $\min\{2s(\psi), 1\}$).

5. Significance and Applications

• Bridging Fields: The work strengthens the connection between Harmonic Analysis (Fourier decay) and Metric Number Theory (Diophantine approximation).
• Normal Numbers: Positive Fourier dimension is a crucial prerequisite for proving the existence of normal numbers within fractal sets (Pollington et al.). This result opens the door to studying normal numbers in more complex, arithmetically constrained sets.
• Multiplicative/Twisted Approximation: The methodology developed (specifically the handling of the Ramanujan sums with general coprime constraints) is suggested to be applicable to multiplicative and weighted Diophantine approximation problems, which are currently open.
• Deterministic Fractals: It contributes to the rare class of explicitly constructed deterministic fractals that are Salem sets, moving beyond random constructions.

In summary, Tan and Zhou have successfully extended the theory of Fourier dimensions in Diophantine approximation to the most general coprime setting, unifying previous homogeneous and inhomogeneous results and settling a significant open conjecture in the field.