{"id":"19c946fc-a75a-4ff7-8434-6c7874c7b7dc","arxiv_id":"2412.12070","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Measures with large Fourier l1 dimension satisfy Khinchin-type and Gallagher-type Diophantine laws, giving new approximation and counting results on missing-digit fractals.","lead":"Mathematicians proved new versions of two classical theorems about how well a typical number can be approximated by fractions, for numbers that live on missing-digit fractals like the Cantor set. The results hold when a Fourier-based measure of the fractal's size is large enough, and they come with new bounds for counting rational points near such fractals.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Divergence half of Theorem 1.8 depends on unproved Lebesgue second-moment estimate (2.5) imported from [25]; full measure requires C arbitrarily close to 1.","rationale":"The reader's weakest-assumption identification is precisely the load-bearing concern I find. Theorem 1.8 is split into convergence (Theorem 1.12) and divergence (Theorem 1.13). The convergence side is proved in this paper with self-contained ETP estimates. The divergence side, however, relies on Lemma 2.4, which requires the Lebesgue second-moment bound (2.5). That bound is quoted from the authors' earlier preprint [25] and not reproved. Since Lemma 2.4 gives full measure only when C can be taken arbitrarily close to 1, any fixed slack in (2.5) reduces the conclusion to a positive lower bound strictly less than 1. I checked the surrounding inequalities: the ETP in the divergence case does follow from the stated Fourier dimension threshold (1.1), and the VTP in Theorem 4.1 is proved in the paper. The only unverified input in the chain is Theorem 2.8. The concern is not about internal inconsistency but about an external dependency. Given that the paper itself is otherwise complete and the imported estimate is plausible from the cited companion work, the appropriate verdict is CONDITIONAL, consistent with the reader's assessment. A self-contained proof or a published reference for Theorem 2.8 would remove the condition.","tokens_in":23117,"tokens_out":8686,"duration_ms":75346,"concrete_test":"Independently verify Theorem 2.8 by re-deriving (2.5) for the two admissible function systems of Examples 2.5 and 2.6 from the Fourier estimates in Sections 2.4 and 4, without invoking reference [25]. In particular, check that the error term in the analogue of [25, Lemma 6.6] is o(E_N(lambda)^2) for the multiplicative system; if the off-diagonal contribution is only O(E_N(lambda)^2) with a positive constant, then C cannot be chosen arbitrarily close to 1 and Lemma 2.4 gives only mu(E_infinity) >= 1/C < 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The divergence half of the main theorem (Theorem 1.8) rests on Theorem 1.13, whose proof uses Lemma 2.4. Lemma 2.4 outputs mu(E_infinity) >= 1/C, so full measure requires C in the Lebesgue second-moment estimate (2.5) to be arbitrarily close to 1. This estimate is imported as Theorem 2.8 from the authors' unpublished preprint [25] (arXiv:2408.10911) and is not proved or even sketched here. For the multiplicative admissible function system in Example 2.6 (divergent Gallagher), the second moment is the delicate part: the smooth functions are sums over roughly m^(k-1) rectangles of different shapes, and the off-diagonal contribution must be o(E_N(lambda)^2) to let C approach 1. If Theorem 2.8 fails for this system, or if the error term in [25, Lemma 6.6] leaves a fixed positive C-1, then the divergent Gallagher conclusion collapses; the same applies to divergent Khinchin via Example 2.5. The convergence half (Theorem 1.12) is self-contained, so the concern is isolated but load-bearing: without (2.5) with C->1, Theorem 1.8 only gives mu(W) >= 1/C, not 1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves metric Diophantine approximation results for Borel probability measures in terms of the Fourier ℓ1 dimension. The main theorem (Theorem 1.8) states that if dim_{ℓ1}(μ) > k - (k-1)/(k+1), then μ is Khinchin, and if, in addition, μ is a product of measures each with dim_{ℓ1} > 1 - 1/(k+1), then μ is Gallagher. The proof develops expectation and variance transference principles: the ETP uses Fourier decay of the measure in dyadic annuli, and the VTP reduces the variance to a counting estimate for solutions of a system of Diophantine equations. The authors then apply these principles to missing-digit fractals, obtaining also results on rational point counting (Theorem 1.17), Hausdorff dimension bounds (Theorem 1.15), and intrinsic approximation (Theorem 1.22).","tokens_in":23320,"tokens_out":11073,"duration_ms":86213,"significance":"If correct, this is the first Fourier-dimension threshold that guarantees both the Khinchin and Gallagher properties for arbitrary measures. For missing-digit fractals, Appendix A shows the thresholds are met for suitable digit sets, e.g., when the base b is large and #D = b^k - 1, so the theorem goes beyond the recent Hausdorff-dimension threshold of Khalil–Luethi. The proofs in Sections 3–5 are detailed, and the final inequality in the VTP reduces exactly to the assumed condition (1.1). The paper also provides a general algorithm to estimate dim_{ℓ1} for missing-digit measures.","major_comments":[{"comment":"The divergent half of the main theorem is not self-contained. Theorem 2.8 quotes the Lebesgue second-moment estimate (2.5) with C = C(k,w) arbitrarily close to 1 from the authors' preprint [25], without proof. This estimate is load-bearing: Lemma 2.4 concludes μ(E∞) ≥ 1/C, and the full-measure statement μ(E∞) = 1 in Theorem 1.13 requires C to be arbitrarily close to 1 for the exact admissible function systems of Examples 2.5 and 2.6. If (2.5) is only available with a fixed C > 1, the divergence conclusions of Theorem 1.8 would reduce to positive-measure results. The authors should either include a proof of (2.5), or state Theorem 1.13 as conditional on the resolution of [25].","section":"Section 2.6, Theorem 2.8"},{"comment":"The paper's advertised applicability to missing-digit fractals rests on the Fourier ℓ1 dimension estimates quoted from the preprints [58, Theorem 2.15] and [23, Theorem 2.6]. These are not reproved in the present paper. Since the examples are used to show that the condition (1.1) is non-vacuous (e.g., Theorem 1.20 and the sentence following Corollary 1.18), the authors should provide the necessary statements in full or prove the needed cases.","section":"Appendix A, Theorem A.1"},{"comment":"In the concluding step of the VTP proof, the notation dim_{ℓ1}(μ) is used as if the defining Fourier estimate holds at the supremum. Since dim_{ℓ1} is a supremum, the argument should fix an exponent κ slightly below dim_{ℓ1}(μ), use the corresponding estimate, and pass to the limit; the strict inequality in (1.1) leaves the required room. This is a presentation gap, but it should be corrected to make the proof fully rigorous.","section":"Section 4.3"}],"minor_comments":[{"comment":"For the convergent Khinchin case, the replacement of ψ by max{ψ, ψ_L} is justified, but the monotonicity of the resulting function should be stated explicitly, since the classical theorems require ψ non-increasing.","section":"Section 5.1"},{"comment":"The statement of Theorem 3.2 uses a fixed d but allows the individual dj to vary within N^{-τ}d and N^{τ}d; the final bound is stated in terms of d. A short note on the uniformity of the implied constants in τ would be helpful.","section":"Section 3.2"},{"comment":"In the lower bound of Theorem 1.17, the single dyadic block [N,2N) with N = floor((Q-1)/2) is used; the wording 'whenever Q is sufficiently large' should clarify that the implied constants may depend on the measure μ through κ and κ2.","section":"Section 5.4"},{"comment":"The inhomogeneous definitions use the same symbol W_k(ψ,y) as the homogeneous case; consider adding a remark that the homogeneous results correspond to y=0.","section":"Page 5, Definition 1.11"},{"comment":"The existence of a missing-digit fractal with dim_H(K) < k - k/(k+1) and D ⊇ {0,...,b-1}^{k-1} × {0} is asserted but not demonstrated; the reader can fill this in by taking #D = b^k - 1 with large b, but a one-line explanation would improve readability.","section":"Example 1.19"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially a companion to the authors' preprint [25], and its main new contribution is the transfer of the moment transference framework to missing-digit measures via the Fourier ℓ1 dimension. The referee recommends that the editors ensure [25] is either published, accepted, or included as an appendix, since the divergence results depend on it. The overall quality is high and the ideas are novel."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sam,\n\nHere's my read of Chow-Yu (2412.12070). The headline: Theorem 1.8 is the first Fourier-l1-dimension criterion I know that gives both Khinchin and Gallagher for arbitrary measures, and the thresholds look right. The paper deserves a serious referee, but the divergence half is conditional on an estimate that is not proved here.\n\nWhat's genuinely new: the ETP and VTP machinery adapted to missing-digit fractals, with explicit thresholds. Theorem 3.1 (dyadic split ETP) and Theorem 4.1 (VTP) are self-contained, and the final inequality in §4.3 rearranges exactly to (1.1). The applications—counting rational points near missing-digit fractals, intrinsic approximation, and the Hausdorff dimension bound—are natural and follow cleanly once the main criteria are in place. The appendix gives concrete digit sets where dim_l1 is close to dim_H, so the main theorem actually applies. This is good mathematical craft, and the self-citations to [23,25,56,58] are procedural tools, not padding.\n\nWhere I worry: the reader's stress-test lands. The divergent Khinchin and Gallagher statements (Theorem 1.13) use Lemma 2.4, which outputs mu(E_infinity) >= 1/C. To get full measure you need C arbitrarily close to 1. That is precisely the content of Theorem 2.8, quoted from the authors' earlier preprint [25] with no proof or even a sketch. Example 2.6 explains why the multiplicative second moment is delicate: the bump functions are sums over roughly m^(k-1) rectangles of different shapes, and the off-diagonal contribution must be o(E_N(lambda)^2). If [25] leaves a fixed positive C-1, Theorem 1.13 only gives positive measure, not 1, and Theorem 1.8 collapses to a one-sided result. This is load-bearing, not cosmetic.\n\nThe convergence half (Theorem 1.12) is self-contained, and the paper's own Appendix A.2 explicitly flags that the upper-bound counterpart is unproved and unnecessary. That is honest and not a problem.\n\nMy recommendation: send it to a strong referee, but insist that the authors either include a self-contained proof of Theorem 2.8 for the admissible function systems used here or replace the reference with a published version. Without that, the divergence theorem is a conditional result. As it stands, I would trust the convergence theory and the counting applications; I would treat the divergent part as conditional.\n\nFor your reading group: worth discussing, especially the second-moment question. I'd cite the convergence results once the divergence issue is resolved.","headline":"A genuinely new Fourier-dimension criterion for Khinchin and Gallagher, with a solid self-contained convergence theory, but the divergent half is conditional on an unproved second-moment estimate imported from the authors' unpublished preprint.","tokens_in":23982,"tokens_out":1871,"would_cite":true,"duration_ms":16303,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","28A80","42B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Fourier dimension threshold makes measures Khinchin and Gallagher.","keywords":["Diophantine approximation","missing-digit fractals","Khinchin's theorem","Gallagher's theorem","Fourier ℓ1 dimension","metric number theory","Littlewood's conjecture","rational points on fractals"],"falsifier":"Compute the optimal constant $C$ in (2.5) for the admissible function systems of Examples 2.5 and 2.6; if the infimum of $C$ over allowed bump functions is strictly greater than 1, then Lemma 2.4 cannot deliver $\\mu(W)=1$ and the divergence conclusions of Theorem 1.13 collapse to a lower bound. Alternatively, search for a Borel probability measure with $\\dim_{\\ell_1}(\\mu) > k - (k-1)/(k+1)$ whose Khinchin divergence set has measure strictly less than 1, which would directly refute Theorem 1.8.","tokens_in":22839,"feed_emoji":"📐","tokens_out":13609,"duration_ms":101885,"temperature":0.7,"pith_summary":"The paper proves that a single number attached to a probability measure — its Fourier $\\ell_1$ dimension, measuring how fast Fourier coefficients decay on average — controls whether the measure inherits the two central theorems of metric Diophantine approximation. If the dimension exceeds $k - (k-1)/(k+1)$, the measure is Khinchin: typical points are approximated by rationals at the same rate as Lebesgue-typical points, for every non-increasing approximation function. If the measure is a product of one-dimensional measures and each factor has dimension above $1 - 1/(k+1)$, it is also Gallagher, so the multiplicative version of the dichotomy holds. These give Fourier-dimension criteria that yield both properties for arbitrary Borel measures, and the appendix shows that natural missing-digit fractals — Cantor-type sets obtained by deleting digits in a large base — meet the thresholds. The reader should care because this settles, for a concrete family of fractal measures, a piece of the conjecture that natural measures behave like Lebesgue measure unless there is an obvious affine obstruction.","feed_headline":"One dimension threshold forces Khinchin and Gallagher laws on fractals","feed_subtitle":"For large-base missing-digit Cantor sets, almost every point approximates rationals as Lebesgue-typical points do.","key_machinery":"The load-bearing objects are the Fourier $\\ell_1$ dimension $\\dim_{\\ell_1}(\\mu)$, defined as the supremum of $s$ such that the $\\ell^1$ sums of Fourier coefficients over boxes of radius $Q$ grow at most like $Q^{k-s}$, and the moment transference principles. The expectation transference principle (ETP) compares $\\mu$-averages of smoothed indicators of the approximation sets with their Lebesgue averages; the variance transference principle (VTP) shows the $\\mu$-variance matches the Lebesgue variance up to a negligible error. The VTP is the heart of the divergence theory: after expanding in Fourier series, the non-diagonal contribution becomes a sum over integer solutions to the linear system $t n + t' n' = x$ with divisibility constraints inherited from the $n^{-1}$-periodicity of the level sets, and counting these solutions with a divisor bound produces an error controlled by $\\dim_{\\ell_1}(\\mu)$. The threshold (1.1) is exactly the condition that this error is smaller than the square of the expectation. The admissible set and function systems decompose the simultaneous and multiplicative approximation sets into unions of rectangles $A_n(d_{i,1},\\ldots,d_{i,k})$ of controlled sizes and volumes, which is what makes the Fourier analysis tractable.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.8: any Borel probability measure $\\mu$ on $\\mathbb{R}^k$ with Fourier $\\ell_1$ dimension $\\dim_{\\ell_1}(\\mu) > k - (k-1)/(k+1)$ is Khinchin — the set of vectors $x$ satisfying $\\max_j \\|n x_j\\| < \\psi(n)$ for infinitely many $n$ has $\\mu$-measure $1$ whenever $\\sum_n \\psi(n)^k = \\infty$ and measure $0$ when this series converges. If in addition $\\mu = \\mu_1 \\times \\cdots \\times \\mu_k$ is split and each factor satisfies $\\dim_{\\ell_1}(\\mu_j) > 1 - 1/(k+1)$, then the same dichotomy holds for the multiplicative condition $\\|n x_1\\|\\cdots\\|n x_k\\| < \\psi(n)$, so $\\mu$ is Gallagher. The proofs establish both properties in the stronger inhomogeneous form with arbitrary shifts $y \\in \\mathbb{R}^k$: a convergence theory requiring only the weaker Fourier condition (1.3), and a divergence theory using (1.1). For missing-digit fractals whose digit set is a Cartesian product or nearly the full digit set, Theorem A.1 shows that $\\dim_{\\ell_1}$ is arbitrarily close to the Hausdorff dimension for large base $b$, so the thresholds are met; the same Fourier machinery yields counting estimates for rational points near such fractals (Theorems 1.17 and 1.20) and a refinement of intrinsic Diophantine approximation (Theorem 1.22).","pith_inferences":["If the thresholds (1.1) and (1.2) are sharp, then the Fourier $\\ell_1$ dimension is the correct single-number statistic for metric Diophantine approximation on self-similar fractals; constructing measures just below the threshold that fail to be Khinchin or Gallagher would be a direct test of that sharpness.","The moment transference machinery should extend to other self-similar measures with rational contraction ratios and the open set condition, where $\\dim_{\\ell_1}$ can be computed by the algorithm in Theorem A.2; one could then try to push the thresholds toward the Hausdorff dimension, which is what the full Dream Theorem would require without the large-base restriction.","The counting results suggest a fractal analogue of dimension growth: for missing-digit fractals that do not contain affine hyperplanes, $\\#N_K(Q,0)$ should be substantially smaller than the trivial $Q^k$ bound; Theorem 1.20 shows the obstruction is real and concentrated on affine-linear structure.","A numerical experiment for $k=2$ could test the Gallagher criterion near the boundary: computing the second-moment constant in (2.5) for truncated digit sets would indicate whether the condition $\\dim_{\\ell_1}(\\mu_j) > 2/3$ is genuinely the right factor-level threshold."],"forward_implications":["Missing-digit measures with a product-like digit set and sufficiently large base $b$ satisfy both the Khinchin and Gallagher laws; for digit sets of size $b^k - 1$ the Fourier $\\ell_1$ dimension exceeds $k - \\varepsilon$ for large $b$, so the main threshold is met.","The inhomogeneous forms hold for every shift $y$: the sets $W_k(\\psi,y)$ and $W^\\times_k(\\psi,y)$ have $\\mu$-measure $0$ or $1$ exactly according to the convergence or divergence of their defining series, so the approximation behaviour is stable under translation of the targets.","For missing-digit fractals satisfying (1.3), the estimate $\\#N_K(Q,\\delta) \\asymp \\delta^{k-\\dim_H(K)} Q^{\\dim_H(K)+1}$ holds for $\\delta \\gg Q^{-\\eta-1/k}$, giving power-saving bounds on rational points on the fractal and sharp bounds in extremal examples.","Almost every point of such a fractal is not intrinsically very well approximable for exponents $\\tau > 1/k - \\varepsilon$ (Theorem 1.22), refining the previous threshold."],"supporting_citations":[{"why":"Supplies the moment transference principles and the Lebesgue second-moment estimate (Theorem 2.8) on which the divergence theory depends.","marker":"[25]"},{"why":"Introduced the use of Fourier $\\ell_1$ dimension for rational-point counting and self-similar sets, providing the dimension statistic the thresholds are stated in.","marker":"[56]"},{"why":"Divergence Borel–Cantelli lemma turns the expectation and variance bounds into full-measure conclusions.","marker":"[13]"},{"why":"Develops the $k=1$ counting and dimension-computation method that Theorem A.2 and the one-dimensional cases build on.","marker":"[23]"},{"why":"Supplies Theorem A.1, the lower bounds on $\\dim_{\\ell_1}$ for product-like missing-digit measures that make the main theorem applicable.","marker":"[58]"},{"why":"Previous result establishing Khinchin's theorem for missing-digit fractals with Hausdorff dimension close to $k$, which the new threshold generalises in the large-base regime.","marker":"[37]"}],"fun_headline_variants":["Large-base fractals: Khinchin and Gallagher via Fourier dimension","Threshold condition makes missing-digit sets Khinchin and Gallagher","Inhomogeneous laws on fractals from a single Fourier dimension bound","Cantor sets meet Khinchin and Gallagher when base is large"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The divergence half of the main theorem rests on a Lebesgue second-moment estimate (Theorem 2.8) quoted without proof from an earlier preprint by the authors, and the proof needs the constant in that estimate to be arbitrarily close to 1 for the specific approximating functions used; if that fails, the full-measure conclusion degrades to $\\mu(W) \\ge 1/C$.","fun_headline_variants_meta":{"raw":{"variants":["Large-base fractals: Khinchin and Gallagher via Fourier dimension","Threshold condition makes missing-digit sets Khinchin and Gallagher","Inhomogeneous laws on fractals from a single Fourier dimension bound","Cantor sets meet Khinchin and Gallagher when base is large"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1699,"prompt_tokens":909,"completion_tokens":790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":717}},"tokens_in":525,"tokens_out":790,"duration_ms":7260,"temperature":1.0,"reasoning_tokens":717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:18:25.079262+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the optimal constant $C$ in (2.5) for the admissible function systems of Examples 2.5 and 2.6; if the infimum of $C$ over allowed bump functions is strictly greater than 1, then Lemma 2.4 cannot deliver $\\mu(W)=1$ and the divergence conclusions of Theorem 1.13 collapse to a lower bound. Alternatively, search for a Borel probability measure with $\\dim_{\\ell_1}(\\mu) > k - (k-1)/(k+1)$ whose Khinchin divergence set has measure strictly less than 1, which would directly refute Theorem 1.8.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Divergence Borel–Cantelli lemma turns the expectation and variance bounds into full-measure conclusions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the $k=1$ counting and dimension-computation method that Theorem A.2 and the one-dimensional cases build on."},{"cited_title":"Khalil and M","cited_arxiv_id":null,"evidence_quote":"Previous result establishing Khinchin's theorem for missing-digit fractals with Hausdorff dimension close to $k$, which the new threshold generalises in the large-base regime."}],"review_version":1}