Published paper
Role in dependence graphs
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statements:
- Lemma 2.1 · Acta Arithmetica pp. 146–148The potential-game strategy partitions dangerous rationals and assigns a common rational line to each block.
- Lemma 2.4 · Acta Arithmetica pp. 149–150The hyperplane-potential deletion family meets the required budget and forces avoidance.
- Theorem 1.1 · Acta Arithmetica p. 145; proof pp. 150–152 is HAW for every nonnegative with .
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Acta Arithmetica version of record
Erez Nesharim and David Simmons. is hyperplane absolute winning. Acta Arithmetica 164 (2014), no. 2, 145–152.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The theorem that is hyperplane absolute winning is correct for every nonnegative with .
Theorem 1.1No issue found
Weighted badly approximable pairs are HAW
VOR p. 145 · Theorem 1.1; proof pp. 150–152
For positive weights, the potential-game construction deletes line neighborhoods covering every dangerous rational rectangle. The endpoint cases reduce to the one-dimensional absolute-winning theorem. FSU's game equivalence then gives HAW.
Lemma 2.1 source locatorCorrect citation after version-number clarification
The imported An result is the 1204.3610 paper, not An13
VOR pp. 146–147 · Lemma 2.1 and explanatory Remark 2.2; reference [An] p. 152
The bibliography identifies arXiv:1204.3610, later published as An16. NS14 calls the source result Lemma 4.2 because it used an older manuscript version. Current arXiv v2 labels the same common-strip statement Corollary 4.2. An13/arXiv:1203.2996 is not this source.
02Proofs2 reported findingsNo proof issues found
The potential-game budget, scale synchronization, and avoidance conclusion are correct.
Proof of Theorem 1.1No issue found in the reviewed proof
The deletion budget converges and every dangerous rational is eventually covered
VOR pp. 150–152
The geometric series over depth stays below the legal potential budget once is chosen large. Lemma 2.1 supplies a single line for each class, and the first relevant scale for any rational point forces the outcome outside its dangerous rectangle. Equation (2.1) then places the outcome in .
Lemma 2.4No issue found in the reviewed proof
Potential winning converts to absolute winning at the required parameters
VOR p. 149 · Lemma 2.4
Lemma 2.4 is the stated specialization of FSU Theorem C.8. Its quantifiers match the use in the proof: establishing hyperplane potential winning for every and is sufficient for HAW.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.