arXiv:2502.04149v3

Rotational beta expansions and Schmidt games

Hajime Kaneko, Jonathan Caalim, Nathaniel Nollen

math.NT11K1611J2568R1537E0511R52

Paper metadata

Original submission: February 6, 2025
Latest revision: July 9, 2026 · version 3
Claimed MSC: 11K16, 11J25, 68R15, 37E05, 11R52
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 33 pages. Published in Acta Mathematica Hungarica. Version 2: Revised in accordance with the referee report. In particular, we have corrected the deficiencies in the statement and proof of Theorem 4.1. Version 3: Corrections emphasizing completeness of metric space used

Abstract

We consider rotational beta expansions in dimensions 1, 2 and 4 and view them as expansions on real numbers, complex numbers, and quaternions, respectively. We give sufficient conditions on the parameters $α, β\in (0,1)$ so that particular cylinder sets arising from the expansions are winning or losing Schmidt $(α,β)$ -game.

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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 15, 2026

01Statements2 reported findingsCorrect

The sufficient winning conditions for real and complex rotational beta expansions and the losing and winning criteria for quaternion expansions are supported. One inequality sign and one endpoint in auxiliary statements require uniquely determined local corrections; under the prompt's typo and minor-formal rules, neither changes the overall correctness status.

Theorems 2.6 and 3.9Correct

Winning criteria in dimensions one and two

Pages 8–10 and 21–23 · Theorems 2.6 and 3.9 · arXiv:2502.04149v3

In dimension one, Lemma 2.1 and Proposition 2.5 give the required uniform spacing from any Bob center to a full cylinder interval, and inequality (2.2) makes Alice's successive radius budget enter that interval. In dimension two, Theorem 3.5 supplies a square digit set and properties $(C_j)$ , so $V_k(\xi;0)$ is tiled by rotated squares with the stated inradius. The same radial worst-case recurrence works in every direction, and the correction appended to version 3 gives the final move that avoids the boundary of the half-open fundamental domain. The numerical hypotheses are exactly the inequalities needed by these strategies.

Theorems 4.1 and 4.3Correct

Quaternion losing and winning criteria

Pages 24–30 · Theorems 4.1 and 4.3 · arXiv:2502.04149v3

For Theorem 4.1, Bob's center formula retains the digits chosen by Alice while returning the tail to a fixed interior ball; the definition of $C_\Omega$ guarantees both legal containment and exclusion of the target block at every stage. For Theorem 4.3 with real quaternion base, the four coordinates are independent real beta expansions. Alice first chooses simultaneous full cylinder intervals, using the factor $1/2$ so the Euclidean center displacement is legal, and then applies the same radial strategy inside their product. This forces the prescribed quaternion digit. The closure correction at the end of version 3 ensures the Schmidt game is played on a complete space without changing the target.

02Proofs3 reported findingsCorrect

The cylinder geometry and Schmidt-game strategies are correct and complete after the version-3 closure correction. A reversed inequality is a harmless typo and a strict upper endpoint is a minor formal correction; the surrounding derivations uniquely determine both repairs.

Proposition 3.6Typo

The criterion for $(C_1)$ has its inequality reversed

Page 18 · Proposition 3.6 · arXiv:2502.04149v3

The statement must read: $(C_1)$ holds if and only if $r\geq\cos\theta+\sin\theta,$ not $r\leq\cos\theta+\sin\theta$ . Indeed, $\xi^{-1}X\subseteq X$ is equivalent to $(\cos\theta+\sin\theta)/(2r)\leq1/2$ . The preceding paragraph already states the correct direction, and $\theta=0$ , $r=2$ gives an immediate check: $\xi^{-1}X=[-1/4,1/4)^2\subset X$ , whereas the printed inequality says the opposite. Reversing the sign is uniquely determined and preserves the following conclusion because every square digit set satisfies $r>(2N-1)(\cos\theta+\sin\theta)\geq\cos\theta+\sin\theta$ .

Full paper, version 3 ↗

Theorem 3.5Minor formal correction

The square-digit upper endpoint should be included

Page 17 · Theorem 3.5, last sentence · arXiv:2502.04149v3

In the claimed equivalence for $(C_2)$ , the upper condition should be $v_N^{(2)}(\theta)<r\leq u_N(\theta),$ rather than $r<u_N(\theta)$ . Proposition 3.3 includes equality at $u_N$ , and the proof of $(C_2)$ depends only on the strict lower inequality $f_N^{(2)}(r)>0$ . For example, at $\theta=0$ , $N=1$ , and $r=u_1(0)=3$ , the digit set is $\{-1,0,1\}^2$ and the translated square $\xi^{-2}X+\xi^{-1}a$ lies in $X$ for every digit $a$ , so $(C_2)$ holds. Theorem 3.8 already prints the corresponding intervals closed at $u_N$ . Changing this one endpoint is local, requires no new argument, and creates no downstream defect.

Full paper, version 3 ↗

Sections 2–4 and CorrectionsCorrect and complete

Cylinder and game arguments

Pages 4–30 and page 33 · central proofs and Corrections · arXiv:2502.04149v3

The admissibility lemmas bound both the spacing and the possible run of shortened cylinders. Conditions $(C_k)$ turn zero-digit cylinders into exact contracted fundamental domains. In each game proof, collinear outward motion maximizes distance from Alice's chosen cylinder center, so the displayed one-dimensional recurrence bounds arbitrary multidimensional play by the triangle inequality. The appended correction replaces each half-open fundamental domain by its closure as the ambient complete metric space and supplies a final interior move, which resolves the only completeness issue without changing any parameter estimate.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2502.04149v3
Authors listed: Hajime Kaneko, Jonathan Caalim, Nathaniel Nollen
Audit date: August 15, 2026

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Rotational beta expansions and Schmidt games

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Winning criteria in dimensions one and two

Quaternion losing and winning criteria

The criterion for (C1)(C_1)(C1​) has its inequality reversed

The square-digit upper endpoint should be included

Cylinder and game arguments

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The criterion for $(C_1)$ has its inequality reversed