MathAudit

Paper metadata

Original submission
January 1, 2024
Latest revision
February 20, 2025 · version 3
Claimed MSC
arXiv categories
math.NT
Comments
Some issues are corrected // 21 pages, 7 figures
Journal reference
Results Math 80, 236 (2025)

Abstract

In this paper we study the classical Schmidt game on two families of sets: one related to frequencies of digits in base-22 expansions, and one connected to the set of the badly approximable numbers. Namely, we describe some nontrivial winning and losing parameters (α,β)(α, β) for these sets.

Dependence graphs

Proof lineage

On Nontrivial Winning and Losing Parameters of Schmidt Games

A statement-restricted graph for arXiv:2401.00614v3. It traces the uniform-distribution, Schmidt-game, Hausdorff-dimension, and Borel-determinacy results actually imported by the v3 proofs. Theorem 2.6 is marked self-contained. Contextual citations, an earlier bound that is not used, and the different determinacy citation added only in the later journal version are excluded.

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AI-generated audit

Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsCorrect

The stated winning and losing regions for digit-frequency sets and the two fixed-constant badly approximable sets are correct. The proof of the losing region for the base-dd set omits the rational-log⁡d(αβ)\log_d(\alpha\beta) case, and an auxiliary scaling lemma has the sign of a logarithm reversed, but the verified repairs below establish the conclusions without changing their hypotheses.

Theorem 2.2Correct

Digit-frequency winning and losing regions

Pages 4–5 and 13–16 · Theorem 2.2 and its proof · arXiv:2401.00614v3

For irrational log⁡m(αβ)\log_m(\alpha\beta), density of the logarithmic orbit lets a player rescale a game interval to any prescribed length window, after correcting the sign in Lemma 3.5. The subsequent block strategies force arbitrarily long biased binary blocks for the losing assertions and give Alice a uniform supply of the required digits for the winning assertions. The quantitative inequalities in the theorem are exactly those needed by those strategies, so the central frequency conclusions follow after the auxiliary repair.

Theorem 2.5Correct

The base-dd fixed-constant badly approximable set is losing in the stated region

Page 7 and pages 16–17 · Theorem 2.5 and its proof · arXiv:2401.00614v3

Put r=αβr=\alpha\beta, γ=1−2α+αβ\gamma=1-2\alpha+\alpha\beta, L=(1−β)−1L=(1-\beta)^{-1}, and U=2c(1−r)/(βγ)U=2c(1-r)/(\beta\gamma). The theorem's hypothesis is U>LU>L. At a trigger time with L<dN∣At∣<UL<d^N|A_t|<U, Bob can center the next ball at a rational m/dNm/d^N; the lower inequality permits that center and the upper inequality, together with the center-control estimate, traps all later play in the corresponding forbidden interval. Such trigger times exist for every rr: an irrational logarithmic step gives a dense orbit, while for a rational step Bob chooses the freely available initial logarithmic phase in the projected interval [log⁡dL,log⁡dU][\log_d L,\log_d U]. This supplies the case omitted from the printed proof and verifies the stated theorem.

Theorem 2.6Correct

The binary fixed-constant badly approximable set is winning in the stated region

Page 7 and pages 17–20 · Theorem 2.6 and its proof · arXiv:2401.00614v3

At each relevant dyadic scale, Alice either keeps the current interval away from the unique forbidden neighborhood or advances to the next scale while preserving the inductive distance bound. The theorem's two parameter inequalities make both alternatives legal. The finite number of smaller scales is immaterial because the target is defined by eventual avoidance. Correcting the two local index and exponent typos identified below makes the invariant agree with the displayed strategy and proves the statement.

02Proofs4 reported findingsContains incorrect or incomplete proofs

Two material steps are not correct as written. Lemma 3.5 asks for a positive fractional part below a negative number, and the proof of Theorem 2.5 uses equidistribution despite allowing rational logarithmic game parameters. Both steps have verified repairs. The other discrepancies are uniquely determined notation errors and do not affect the paper's overall correctness.

Lemma 3.5Incorrect as written · verified repair

The logarithmic rescaling inequality has a reversed sign

Pages 9–10 · Lemma 3.5 and proof · arXiv:2401.00614v3

Because 0<αβ<10<\alpha\beta<1, the printed quantity log⁡αβ(1+ε)\log_{\alpha\beta}(1+\varepsilon) is negative, so the required inequality for a positive fractional part is impossible. Set δ=−log⁡αβ(1+ε)>0\delta=-\log_{\alpha\beta}(1+\varepsilon)>0, ξ=−log⁡αβm\xi=-\log_{\alpha\beta}m, and χ=log⁡αβ(L/l)\chi=\log_{\alpha\beta}(L/l). Density of the orbit of ξ\xi modulo one gives kk with {kξ+χ}<δ\{k\xi+\chi\}<\delta. Taking n=⌊kξ+χ⌋n=\lfloor k\xi+\chi\rfloor then gives Lmk≤(αβ)nl<(1+ε)Lmk,\frac{L}{m^k}\leq(\alpha\beta)^n l<(1+\varepsilon)\frac{L}{m^k}, which is the exact rescaling conclusion used later.

Proof of Theorem 2.5Incomplete as written · verified repair

The rational logarithmic-step case is omitted

Pages 16–17 · proof of Theorem 2.5 · arXiv:2401.00614v3

The proof invokes equidistribution of the sequence of logarithmic interval lengths, but the theorem does not assume log⁡d(αβ)\log_d(\alpha\beta) irrational. The repair is to use the freedom in Bob's initial interval length. If the logarithmic step is irrational, the printed density argument applies. If it is rational, Bob chooses the initial phase in the nonempty projected interval [log⁡dL,log⁡dU][\log_d L,\log_d U] modulo one. The phase then returns periodically, producing infinitely many scales NN with L<dN∣At∣<UL<d^N|A_t|<U. The same center choice and trapping argument finish the proof. The printed closed-limsup-to-open-limsup inclusion is also reversed; the direct trapping argument proves open membership and needs no such inclusion.

Proof of Theorem 2.6Typo

A dyadic distance and one induction index are mistyped

Page 19 · proof of Theorem 2.6 · arXiv:2401.00614v3

When Bob's interval avoids the forbidden neighborhood at scale qiq_i, the directly obtained distance is c/2qi−1c/2^{q_i-1}, not c/2qi+1c/2^{q_i+1}. In the next induction transition, the new error variable is ei+1e_{i+1} rather than eie_i. The surrounding inequalities use the corrected stronger distance and the next index, so these are uniquely repairable typographical errors.

Auxiliary game notationTypo

Several local symbols have unique contextual corrections

Pages 4 and 9–16 · theorem displays and symmetry lemmas · arXiv:2401.00614v3

The statement of Theorem 2.2 omits the backslash in one occurrence of α\alpha; the invariant in Lemma 3.3 should read mkS=mk+S=Sm^kS=m^k+S=S; and a symmetry sentence reverses membership when passing between SS and −S-S. Each intended symbol is fixed by the adjacent definition and none changes a parameter range or a strategic step.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2401.00614v3
Authors listed
Vasiliy Neckrasov, Eric Zhan
Audit date
August 19, 2026
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