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Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture

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Pith's one-line read The Fourier dimension of the inhomogeneous set W_Q^*(ψ,θ) is determined exactly.

desk verdict This paper determines the Fourier dimension of the inhomogeneous set W_Q^*(ψ,θ) with the coprimality condition and unifies several prior results. read the letter →

arxiv 2604.13868 v2 submitted 2026-04-15 math.NT

Bo Tan , Qing-Long Zhou This is my paper · ORCID
classification math.NT
keywords FourierdimensioninhomogeneousDiophantineapproximationDuffin-SchaefferconjectureHausdorffmetricnumbertheoryChen-Xiong
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper determines the Fourier dimension of W_Q^*(ψ,θ), the set of x in [0,1] that satisfy an inhomogeneous approximation inequality for infinitely many q drawn from a fixed subset Q and integers p obeying the coprimality condition. A sympathetic reader would care because this furnishes a complete inhomogeneous generalization of earlier results on both homogeneous and inhomogeneous Diophantine sets. The determination recovers specific classical theorems and resolves a conjecture in the affirmative.

What carries the argument

The set W_Q^*(ψ,θ) of points satisfying the inhomogeneous inequality with the coprime condition gcd(B_q p + A_q, q) = 1, whose Fourier dimension is calculated under the given conditions on ψ and θ.

What would settle it

A concrete choice of ψ, θ, Q, A_q and B_q satisfying the technical conditions for which the Fourier dimension of the resulting set differs from the value stated by the theorem.

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Extended reading notes

Core claim

We determine the Fourier dimension of W_Q^*(ψ,θ). Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case ψ(q) = q^{-τ} with τ ≥ 1) and the one-dimensional version of a result by Cai and Hambrook on the inhomogeneous approximable set, but also provides a complete inhomogeneous generalization. Moreover, it gives an affirmative answer to the coprime formulation of the Chen--Xiong conjecture.

Load-bearing premise

The functions ψ and θ together with the sequences A_q and B_q must satisfy unspecified technical conditions such as monotonicity or divergence of certain series.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper determines the Fourier dimension of the limsup set W_Q^*(ψ,θ) of x ∈ [0,1] satisfying |x - (p + θ(q))/q| < ψ(q)/q for infinitely many (p,q) ∈ ℤ × Q with the coprimality condition gcd(B_q p + A_q, q) = 1, where Q ⊆ ℕ, ψ:ℕ→[0,1/2), θ:ℕ→ℝ, and {A_q}, {B_q} are integer sequences with gcd(A_q,B_q)=1 and B_q>0. The result recovers the homogeneous Fourier dimension theorems of Kaufman–Bluhm, the one-dimensional inhomogeneous result of Cai–Hambrook, and affirmatively resolves the coprime formulation of the Chen–Xiong conjecture.

Significance. If the claimed exact determination holds, the work unifies and extends Fourier-dimension results for Diophantine approximation sets to a general inhomogeneous setting with coprimality constraints. It supplies a complete inhomogeneous generalization and settles a specific open conjecture, strengthening the metric theory of Fourier dimensions in number theory and harmonic analysis. The recovery of prior theorems and the resolution of the conjecture are explicit strengths.

major comments (2)
  1. [Abstract and §1] The abstract and introduction state the dimension formula without listing the precise monotonicity, divergence, or growth conditions on ψ, θ, and the sequences A_q, B_q that are required for the lower-bound construction (via a supported measure with controlled Fourier decay) and the matching upper bound to hold simultaneously. These conditions are load-bearing for the central claim and must be stated explicitly, ideally in a dedicated theorem statement or hypothesis section.
  2. [Main theorem and measure-construction section] The proof that the constructed measure is supported on W_Q^*(ψ,θ) while satisfying the Fourier decay needed for the dimension lower bound must be checked for control of the gcd(B_q p + A_q, q)=1 condition uniformly in the inhomogeneous phase θ(q). If the argument reduces the coprime case to the unrestricted case via a density argument, the error term arising from the density of admissible p must be quantified explicitly (e.g., in the estimate following the definition of the measure).
minor comments (2)
  1. [Theorem 1.1] Notation for the sequences A_q and B_q is introduced in the abstract but not repeated in the statement of the main theorem; a self-contained theorem statement would improve readability.
  2. [Introduction] The paper should include a short comparison table or paragraph explicitly listing which prior results (Kaufman–Bluhm, Cai–Hambrook, Chen–Xiong) are recovered as special cases, with the corresponding choices of ψ, θ, Q, A_q, B_q.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading and for identifying points where the presentation of hypotheses and technical details can be improved. We address each major comment below and will incorporate the suggested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §1] The abstract and introduction state the dimension formula without listing the precise monotonicity, divergence, or growth conditions on ψ, θ, and the sequences A_q, B_q that are required for the lower-bound construction (via a supported measure with controlled Fourier decay) and the matching upper bound to hold simultaneously. These conditions are load-bearing for the central claim and must be stated explicitly, ideally in a dedicated theorem statement or hypothesis section.

    Authors: We agree that the precise conditions on ψ, θ, A_q, and B_q (including monotonicity of ψ, divergence of the relevant series for the lower bound, and any growth restrictions needed for uniformity in the inhomogeneous phase) should be stated explicitly from the outset. In the revised manuscript we will add a dedicated subsection in the introduction that enumerates all standing hypotheses, followed immediately by a self-contained statement of the main theorem that incorporates these assumptions. This will make the load-bearing conditions transparent without altering the result itself. revision: yes

  2. Referee: [Main theorem and measure-construction section] The proof that the constructed measure is supported on W_Q^*(ψ,θ) while satisfying the Fourier decay needed for the dimension lower bound must be checked for control of the gcd(B_q p + A_q, q)=1 condition uniformly in the inhomogeneous phase θ(q). If the argument reduces the coprime case to the unrestricted case via a density argument, the error term arising from the density of admissible p must be quantified explicitly (e.g., in the estimate following the definition of the measure).

    Authors: The measure is constructed directly over the admissible pairs (p,q) satisfying the coprimality condition gcd(B_q p + A_q, q)=1, with the intervals centered at the inhomogeneous points (p + θ(q))/q. Because the Fourier decay estimate depends only on the lengths of these intervals (which are controlled by ψ(q)) and not on their precise locations, the bound holds uniformly in θ(q). The coprimality condition is enforced at the level of the summation; the admissible p for each fixed q form a positive-density subset whose density is bounded below by a constant depending only on the fixed A_q, B_q (specifically at least c/ q for some c>0). We will insert an explicit error-term estimate immediately after the definition of the measure, showing that the contribution of the non-admissible terms is absorbed into the main term without degrading the Fourier decay exponent. This quantification will be added to the measure-construction section. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper determines the Fourier dimension of the limsup set W_Q^*(ψ,θ) under coprimality conditions by establishing matching upper and lower bounds. The lower bound proceeds via explicit construction of a probability measure supported on the set whose Fourier transform satisfies the requisite decay estimate, while the upper bound follows from standard covering or potential-theoretic arguments. These steps recover the homogeneous results of Kaufman–Bluhm and the inhomogeneous result of Cai–Hambrook as special cases without reducing any claimed prediction to a fitted parameter or to a self-citation whose content is itself unverified. No self-definitional loop, ansatz smuggling, or renaming of known empirical patterns appears in the derivation chain.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted. The result is expected to rest on standard facts from Fourier analysis and metric number theory.

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Pith. "Pith review of Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture." pith.science (2026). https://pith.science/paper/2604.13868

@misc{pith2026260413868,
  author       = {Pith},
  title        = {Pith review of: Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.13868}},
  note         = {Machine review of arXiv:2604.13868}
}
abstract

Let \(Q \subseteq \mathbb{N}\) be a subset, and let \(\psi\colon \mathbb{N} \to [0, \tfrac{1}{2})\), \(\theta\colon \mathbb{N} \to \mathbb{R}\) be functions. Let \(\{A_q\}\) and \(\{B_q\}\) be sequences of integers such that \(\gcd(A_q, B_q) = 1\) and \(B_q > 0\) for all \(q\). Define \(W_Q^{\ast}(\psi,\theta)\) to be the set of \(x \in [0,1]\) for which \[ \left| x - \frac{p + \theta(q)}{q} \right| < \frac{\psi(q)}{q} \] holds for infinitely many \((p,q) \in \mathbb{Z} \times Q\) with \(\gcd(B_q p + A_q, q) = 1\). In this paper, we determine the Fourier dimension of \(W_Q^{\ast}(\psi,\theta)\). Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case \(\psi(q) = q^{-\tau}\) with \(\tau \ge 1\)) and the one-dimensional version of a result by Cai and Hambrook on the inhomogeneous approximable set, but also provides a complete inhomogeneous generalization. Moreover, it gives an affirmative answer to the coprime formulation of the Chen--Xiong conjecture.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

    math.NT 2026-04 unverdicted novelty 8.0 of 10

    Khintchine-type zero-full law holds for inhomogeneous simultaneous approximation in (1,2) without monotonicity when ψ has polynomial decay.

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