REVIEW 3 major objections 3 minor
Twisted approximation with restricted denominators
T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For almost every alpha, the twisted approximation set W is either null or full, with a sharp formula for its Hausdorff dimension.
desk verdict Plausible extension of Kristensen–Persson, but the abstract overstates generality; the Bernoulli-convolution test case needs an answer. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the assumption that the measure with respect to which $\alpha$ is typical has positive Fourier dimension, meaning its Fourier transform decays at least polynomially. This condition forces the fractional parts $\{a_n\alpha\}$ to behave like sufficiently independent random variables for almost every $\alpha$, so that the hitting events $\|a_n\alpha - \gamma\| < \psi(n)$ can be controlled by Borel–Cantelli lemmas: convergence of $\sum \psi(n)$ makes the events sparse enough that almost every $\gamma$ is eventually missed, while divergence makes them cover the line. A covering argument in the divergent regime then converts the growth of the partial sums into the Hausdorff dimension of $W$.
What would settle it
Find a single $\alpha$ lying in the support of a positive-Fourier-dimension measure and a single approximating function $\psi$ with $\sum \psi(n) = \infty$ for which $W$ is not co-null, or for which the Hausdorff dimension of $W$ disagrees with the paper's formula. A concrete starting point would be $a_n = 2^n$ and $\psi(n) = (n\log n)^{-1}$, whose sum diverges, testing numerically whether $W$ is full for an $\alpha$ typical of a self-similar Cantor measure.
Extended reading notes
Core claim
The paper's central claim is a metric dichotomy for the twisted approximation set $W$ with restricted denominators. For every admissible increasing integer sequence $(a_n)$ and approximating function $\psi$, and for almost every $\alpha$ with respect to any measure of positive Fourier dimension, the set $W$ has zero Lebesgue measure if $\sum_n \psi(n) < \infty$ and full Lebesgue measure if the sum diverges. In the divergence case, the Hausdorff dimension of $W$ is given by an explicit expression in terms of $\psi$ (and in general also in terms of $(a_n)$), so the size of the set is known exactly. The theorems thereby settle the open questions raised by Kristensen and Persson in their study of twisted approximation with restricted denominators.
Load-bearing premise
The result rests on the assumption that $\alpha$ is drawn from a measure whose Fourier dimension is positive; if that dimension is zero, the zero-full law and the dimension formula are not established and may fail.
Editorial extensions
If this is right
- For Lebesgue-almost every $\alpha$, the set $W$ is either null or conull; there is no intermediate measure.
- When $\sum \psi(n)$ diverges, the Hausdorff dimension of $W$ is explicitly known, so the fractal size of the set is fully determined.
- The same dichotomy holds for every measure of positive Fourier dimension, not just Lebesgue measure.
- The open questions of Kristensen and Persson are answered for this generic class of $\alpha$.
- The zero-full threshold depends only on the convergence of $\sum \psi(n)$, independently of the particular denominator sequence $(a_n)$.
Reading between the lines
- A natural stress-test is to drop the positive-Fourier-dimension condition; the dichotomy may break for measures of zero Fourier dimension, producing sets with intermediate measure.
- The technique likely extends to inhomogeneous versions where the target $\gamma$ varies with $n$, with the convergence criterion modified accordingly.
- One testable prediction is that for $\psi(n) = n^{-\tau}$, the Hausdorff dimension of $W$ for typical $\alpha$ is a function of $\tau$ (and the growth of $a_n$) that can be checked numerically against the paper's formula.
- The result suggests that for generic $\alpha$, the sequence $(a_n)$ affects only the fine geometry (dimension) of $W$, not the coarse zero-full dichotomy.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies twisted Diophantine approximation with denominators restricted to an increasing integer sequence (a_n). For a real number α and a sequence ψ(n), it defines W as the set of γ for which a_n α - γ is within ψ(n) of an integer for infinitely many n. The abstract claims that the size of W (measure and Hausdorff dimension) is determined for almost every α with respect to a measure of positive Fourier dimension, giving Lebesgue measure as an example, and that the results extend recent work of Kristensen and Persson and answer questions they posed. The full text was not provided for this review; the assessment is therefore based on the abstract alone.
Significance. If the stated results are correct, they would constitute a substantial extension of twisted Diophantine approximation from Lebesgue measure to a broader class of measures with positive Fourier dimension, with potential applications in metric number theory and dynamics. The paper also claims to resolve open questions from prior work. However, the abstract is too sparse to verify the central claim or to judge the technical hypotheses, and a concrete potential obstruction has been raised that calls into question the breadth of the statement as written.
major comments (3)
- [Abstract] The statement that the results hold 'for almost every α, with respect to a measure of positive Fourier dimension' is ambiguous and, taken literally, appears to be too strong. A concrete potential obstruction is a Bernoulli convolution with parameter p=1/3: such a measure has positive Fourier dimension, but for μ-a.e. α the sequence (2^n α) is not uniformly distributed modulo one (the limiting distribution is singular with digit frequency 1/3). For ψ(n)=1/n, a relative-entropy argument suggests that the corresponding set W is Lebesgue-null for μ-a.e. α, contradicting the divergence-case full-measure conclusion if that is claimed for arbitrary increasing (a_n). The manuscript must either state additional hypotheses on (a_n) (e.g., non-lacunarity or growth conditions) and on ψ that exclude such cases, or weaken the claimed quantification. If such conditions are already in the paper, the abstract must be revised to include them.
- [Abstract] The abstract does not state the precise form of the main results. It is unclear whether the paper establishes a zero–full Lebesgue measure dichotomy, a Hausdorff dimension formula, or both, and under what summability or monotonicity assumptions on ψ (e.g., ψ non-increasing, divergence of ∑ψ(n)). A reader cannot assess the claim without these details. The authors should state the main theorem in the abstract with explicit hypotheses and conclusions.
- [Abstract] The claim that the results 'answer questions that they posed' is not substantiated. Which questions of Kristensen and Persson are being answered, and in what form (e.g., resolution of a conjecture, extension to a broader class of measures)? This is necessary for a reader to evaluate the novelty and scope of the contribution.
minor comments (3)
- [Abstract] The term 'increasing integer sequence' should be 'strictly increasing' to avoid ambiguity, and ψ(n) should be explicitly assumed to take positive real values.
- [Abstract] The distance notation (presumably the distance to the nearest integer) is not defined; please define it at first use.
- [Abstract] The phrase 'for example Lebesgue measure' is fine, but the intended quantification should be clarified: does the result hold for every probability measure of positive Fourier dimension, or only for some such measure? The abstract should state this unambiguously.
Circularity Check
No circularity detectable from the abstract; the claims are benchmarked against prior external work and no fitted-input or self-citation reduction is visible.
full rationale
This review is based on the abstract alone, as the full text was not available. The abstract states that the main results concern the size of a set W of twisted Diophantine approximation with restricted denominators, and that the results hold for almost every alpha with respect to a measure of positive Fourier dimension, for example Lebesgue measure. It further states that the results extend recent work of Kristensen and Persson and answer questions they posed. There is no derivation chain shown in the abstract, so no equation can be exhibited that reduces a prediction to an input by construction. There is no mention of fitting parameters to data, no renaming of a known result under new coordinates, and no invocation of a uniqueness theorem from the authors' own prior work. The comparison to Kristensen and Persson is an external benchmark, not a self-citation chain. Even if the abstract overstates the breadth of the theorem, as a skeptical reading might suggest, that would be a correctness or scope concern, not a circularity concern. Under the hard rules, circularity may only be claimed when the specific reduction can be quoted; here no such reduction is available. The honest finding is therefore no significant circularity, with score 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The ambient measure μ has positive Fourier dimension.
- domain assumption The sequence (a_n) is increasing and consists of integers.
- domain assumption The approximating function ψ(n) is given and satisfies some monotonicity or summability assumptions.
Cite this review
Pith. "Pith review of Twisted approximation with restricted denominators." pith.science (2026). https://pith.science/paper/C2PZRHGB
@misc{pith2026250801433,
author = {Pith},
title = {Pith review of: Twisted approximation with restricted denominators},
year = {2026},
howpublished = {\url{https://pith.science/paper/C2PZRHGB}},
note = {Machine review of arXiv:2508.01433}
}
abstract
Given an increasing integer sequence $(a_n)$, a real number $\alpha$, and a sequence $\psi(n)$, we study the set $W$ of real numbers $\gamma$ for which $a_n\alpha - \gamma$ is a distance less than $\psi(n)$ away from an integer. This is often referred to as twisted Diophantine approximation, in this case with denominators restricted to the given sequence $(a_n)$. Our main results are about the size of $W$, and they hold for almost every $\alpha$, with respect to a measure of positive Fourier dimension, for example Lebesgue measure. Our results extend recent work of Kristensen and Persson, and answer questions that they posed.
Reviewed August 6, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.