{"id":"21c07cc3-c5ec-43c4-86e2-0e7a54ba3d38","arxiv_id":"2508.01433","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper determines the measure and Hausdorff dimension of the set of γ for which a_n α is within ψ(n) of an integer infinitely often, for almost every α with respect to a measure of positive Fourier dimension.","lead":"This mathematics paper studies how close multiples of a fixed number can come to an integer for infinitely many allowed denominator values, and it describes the size of the set of target numbers for which this occurs. The results are proven for almost every choice of the base number under a technical condition, and they extend and answer questions from recent work by Kristensen and Persson.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Positive Fourier dimension alone may be insufficient for arbitrary increasing (a_n): Bernoulli-measure examples suggest the abstract's 'almost every α' claim is overbroad.","rationale":"The reader correctly noted that the abstract omits essential hypotheses on ψ and (a_n). My stress-test sharpens this into a concrete correctness risk: the 'positive Fourier dimension' condition on the parameter measure is not by itself enough to ensure the divergence-case dichotomy for arbitrary increasing integer sequences. The Bernoulli convolution example shows a measure of positive Fourier dimension for which μ-a.e. α has a lacunary orbit that is not Lebesgue-uniformly distributed; the limsup set W then has Lebesgue measure zero even though the radius sum diverges. This is a genuine mathematical obstruction, not a stylistic concern. However, because only the abstract is available, I cannot determine whether the paper's actual theorem includes the counterexample or avoids it through stated hypotheses. The reader's UNVERDICTED verdict remains appropriate: the paper cannot be evaluated without the full statement. If the full text indeed asserts the theorem for all positive-Fourier-dimension measures and all increasing (a_n), the theorem is false and the verdict would need to move to REJECT; if it restricts the measure or the sequence in a way that excludes the Bernoulli example, the concern becomes a demand to disclose the missing condition in the abstract. Thus the verdict is unchanged pending inspection of the full text.","tokens_in":634,"tokens_out":33383,"duration_ms":423454,"concrete_test":"Check the full statement of the main theorem. If it claims the result for every measure of positive Fourier dimension and every increasing integer sequence (a_n), run this test: take μ = Bernoulli convolution with p=1/3, a_n = 2^n, and ψ(n) = 1/n. Compute (analytically or by simulating a μ-typical α) the Lebesgue measure of W∩[0,1]. If it is not 1, the theorem as stated is false; if the full text excludes lacunary sequences or singular measures of this type, the abstract is materially incomplete and must state the restriction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The divergence-case dichotomy (W has full Lebesgue measure when the radii sum diverges) requires that, for μ-a.e. α, the orbit (a_n α) is sufficiently uniformly distributed with respect to Lebesgue. Positive Fourier dimension of μ does not guarantee this for every increasing integer sequence. A concrete obstruction: take μ to be a Bernoulli convolution with p=1/3. Such μ has positive Fourier dimension (the Fourier transform decays as |t|^{-η} for some η>0). For μ-a.e. α, the binary expansion has digit frequency 1/3, so the sequence (2^n α) is not uniformly distributed mod 1; its limiting distribution is a singular measure ν. Choose ψ(n)=1/n, which is non-increasing with divergent sum. For Lebesgue-a.e. γ, a standard relative-entropy calculation gives the expected number of n with |2^n α - γ| < 1/n as ∑ n^{-(H(1/2)+D(1/2||1/3))} = ∑ n^{-1.085}, which converges. Thus W is Lebesgue-null for μ-a.e. α, contradicting the claimed full-measure statement. The abstract must therefore hide an additional condition—for example, that (a_n) is not lacunary, or that μ is absolutely continuous, or that ψ decays fast enough. Without such a condition, the central claim is not merely under-specified but likely false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":904,"tokens_out":4766,"duration_ms":57647,"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":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"minor_comments":[{"comment":"The term 'increasing integer sequence' should be 'strictly increasing' to avoid ambiguity, and ψ(n) should be explicitly assumed to take positive real values.","section":"Abstract"},{"comment":"The distance notation (presumably the distance to the nearest integer) is not defined; please define it at first use.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The review is based on the abstract alone, as the full text was not available. The potential counterexample involving Bernoulli convolutions is serious and should be addressed directly by the authors, either by adding hypotheses or by adjusting the claimed scope. I recommend asking the authors to provide a precise statement of the main theorems, including all conditions on (a_n), ψ, and the measure, and to verify the claims against the obstruction described."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: plausible extension of Kristensen–Persson, but the abstract as written is either overbroad or missing hypotheses. The Bernoulli-convolution counterexample from the stress test is the right thing to check, and the authors need to either add conditions on (a_n) and ψ or the theorem is false.\n\nWhat's genuinely new: the paper answers open questions from Kristensen and Persson, and the use of positive Fourier dimension is a reasonable weakening of Lebesgue measure. If the main dichotomy survives contact with the right hypotheses, this is a useful contribution to metric twisted Diophantine approximation.\n\nThe soft spot: the abstract claims the results hold for almost every α with respect to a measure of positive Fourier dimension, for an arbitrary increasing sequence (a_n). That is not plausible on its face. Take μ a Bernoulli convolution with p=1/3 and a_n=2^n. μ has positive Fourier dimension, but for μ-a.e. α the fractional parts of 2^n α are not equidistributed. With ψ(n)=1/n, a standard entropy calculation suggests the twisted approximation set has Lebesgue measure zero, not full. If the theorem actually proves full Lebesgue measure for all such μ and all increasing (a_n), the proof has a bug. More likely, there are hidden conditions (e.g., (a_n) not too lacunary, or a different quantification over α). The abstract must state them.\n\nI can't verify any of this from the abstract alone. The stress-test example might not land if the theorem is only for a specific μ chosen by the authors, or if the divergence case is stated differently. But it is exactly the kind of edge case a referee should probe.\n\nI don't see any obvious issue with the citation pattern from the abstract; the authors are correctly situating themselves against Kristensen–Persson.\n\nVerdict: worth a serious referee. If the hypotheses are right, the paper is solid, though incremental. If they aren't, a referee will find the counterexample. Either way, send it out.","headline":"Plausible extension of Kristensen–Persson, but the abstract overstates generality; the Bernoulli-convolution test case needs an answer.","tokens_in":1346,"tokens_out":3246,"would_cite":false,"duration_ms":37333,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","11K60","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"For almost every alpha, the twisted approximation set W is either null or full, with a sharp formula for its Hausdorff dimension.","keywords":["twisted Diophantine approximation","restricted denominators","metric Diophantine approximation","Hausdorff dimension","positive Fourier dimension","zero-one law","Khintchine-type theorem"],"falsifier":"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.","tokens_in":465,"feed_emoji":"🎯","tokens_out":9986,"duration_ms":112047,"temperature":0.7,"pith_summary":"The paper studies twisted Diophantine approximation when the denominators are restricted to a fixed increasing sequence $(a_n)$. For $\\alpha$ lying in the support of a measure of positive Fourier dimension — Lebesgue measure is the leading example — it claims that the set $W$ of real numbers $\\gamma$ for which $\\|a_n\\alpha - \\gamma\\| < \\psi(n)$ for infinitely many $n$ obeys a clean zero-or-full law. The Lebesgue measure of $W$ is zero when $\\sum_n \\psi(n)$ converges and full when the sum diverges, and in the divergent case the Hausdorff dimension of $W$ is exactly determined by the decay of $\\psi$ and the growth of $(a_n)$. This extends recent work of Kristensen and Persson and answers questions they left open, giving number theorists a practical criterion for the size of these exceptional sets.","feed_headline":"Twisted Diophantine sets are zero or full for almost every alpha","feed_subtitle":"For almost every alpha, one convergence sum decides both the measure and the dimension of these approximation sets.","key_machinery":"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 $W$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $(a_n)$."],"supporting_citations":[],"fun_headline_variants":["Zero or full: twisted approximation sets for almost every alpha","Sum convergence decides measure and dimension of twisted sets","Twisted approximation: one sum decides zero or full","For almost every alpha, twisted sets are zero or full"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero or full: twisted approximation sets for almost every alpha","Sum convergence decides measure and dimension of twisted sets","Twisted approximation: one sum decides zero or full","For almost every alpha, twisted sets are zero or full"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000634,"raw_usage":{"total_tokens":2855,"prompt_tokens":808,"completion_tokens":2047,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":1983}},"tokens_in":424,"tokens_out":2047,"duration_ms":19184,"temperature":1.0,"reasoning_tokens":1983,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:35:27.479094+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}