arXiv:2608.24401v2
Winning property of counterexamples to Uniform Littlewood's Conjecture
math.NT11J8311J13
Abstract
In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in Bandi-Fregoli-Kleinbock, that is, the set of pairs of real numbers satisfying is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs satisfying is hyperplane absolute winning if for some . In particular, the above sets have full Hausdorff dimension in . In addition, we prove that these sets are absolute winning on every regular planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.
Dependence graphs
Proof lineage
Winning property of counterexamples to Uniform Littlewood's Conjecture
A full recursive graph for the proof-critical sources behind arXiv:2608.24401v2: the ambient and fixed-sequence HAW theorems, curve and line restrictions, dynamical formulations, and the two active appendix examples. Contextual citations and the inactive source block are excluded; exact-source limitations and terminal books are shown on the affected branches.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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
01Statements4 reported findingsNo statement issues found
The ambient hyperplane-absolute-winning theorem, its fixed-sequence refinement, the curve and line restrictions, the homogeneous-dynamics formulations, and the two active appendix examples have compatible hypotheses and conclusions in arXiv:2608.24401v2. No false, ill-posed, or materially overstated central statement was found.
Theorem 1.2 and Corollaries 1.3–1.4No issue found
The ambient winning and full-dimension conclusions follow
PDF pp. 2–3 and 9–22 · Theorem 1.2, Corollaries 1.3–1.4, and §§3–5 · arXiv:2608.24401v2
Failure of the target inequality is covered by the dangerous rational rectangles of Lemma 3.1. The weighted attached-line and simplex lemmas concentrate every same-scale family in controlled affine-line neighborhoods, Propositions 4.1–4.2 keep their total potential within Alice's allowance, and the stage strategy eventually treats every dangerous family. Hyperplane-potential winning therefore gives HAW for . The zero-dimensional ordinary Littlewood exceptional set and countable-intersection stability then give both corollaries, including endpoint weights.
Theorem 1.6No issue found
Power-rapid fixed sequences give HAW intermediate sets
PDF pp. 3–4 and 22–25 · Theorem 1.6 and §6 · arXiv:2608.24401v2
The structured cubic-gap strategy treats . If with , choosing with makes each residue-class subsequence cubic-gap, and is exactly the finite intersection of the corresponding residue-class targets. HAW stability under finite intersections proves the theorem with the stated quantifiers.
Theorems 1.7–1.8No issue found
The curve and line restrictions have the claimed winning behavior
PDF pp. 4 and 25–29 · Theorems 1.7–1.8 and §7 · arXiv:2608.24401v2
On coordinate arcs, structured ambient deletions pull back to legal interval deletions unless the outcome is already outside the auxiliary dual badly approximable set. That auxiliary set is HAW by the An–Beresnevich–Velani curve theorem and the one-dimensional winning equivalence; closed derivative-exception sets are removed using Lemma 2.3. For lines, the printed exponent is exactly the case of the cited line theorem. On rational lines, convergent denominators make the relevant product tend to zero, proving the empty-intersection branch.
Theorems 1.9 and 1.11, Corollary 1.10, and Appendix ANo issue found
The dynamical translations and independence examples have the claimed scope
PDF pp. 5 and 30–35 · §1.4 and Appendix A · arXiv:2608.24401v2
The standard unipotent chart translates the ambient, weighted-ray, and curve results to the stated torus. In Appendix A, the complement of the one-dimensional Sierpiński gasket is not ambient HAW but is HAW on every curve in the theorem's class, while the pullback of under is ambient HAW and misses a nonzero-curvature level curve. These give the two advertised directions of independence.
02Proofs6 reported findingsNo proof issues found
All central internal arguments close in the current version. The audit checked the dangerous-set decomposition, multiweight determinant bounds, both potential estimates, trigger-time strategy, cubic and residue-class deductions, the full curve-transfer chain, the two line cases, and both active appendix constructions. No proof gap requiring a repair was found.
Lemmas 3.1–3.6No issue found in the reviewed proof
The dangerous-set decomposition and simplex bounds preserve the required inequalities
PDF pp. 9–14 · §3 · arXiv:2608.24401v2
The product inequality is divided into the interior weighted rectangles and the two coordinate strips, with the cutoff term handled separately. Primitive attached lines have the printed coefficient bounds. For rational points in one scale cell, the relevant incidence determinant is an integer of absolute value less than one and hence vanishes; weight symmetry and the strip cross-product argument cover the remaining cases. The resulting one- or two-line neighborhoods have exactly the radii used later.
Propositions 4.1–4.2 and §5No issue found in the reviewed proof
The potential budgets and stage strategy are legal
PDF pp. 15–22 · §§4–5 · arXiv:2608.24401v2
The denominator and height partitions cover every dangerous set and have finite active cells. The line-neighborhood radii form geometric sums with the required decay, and the choices of , , and the cutoffs put each contribution inside Alice's budget. The trigger index is unique at a turn, all finite-height families are eventually triggered when Bob's radii shrink, and nesting transports each deletion to the stage endpoint. Thus every non-default outcome lies in .
Lemma 6.2 and proof of Theorem 1.6No issue found in the reviewed proof
The structured strategy retains the coefficient control needed on curves
PDF pp. 23–25 · Lemma 6.2 and proof of Theorem 1.6 · arXiv:2608.24401v2
The strategy makes arbitrarily small by choosing the auxiliary parameter before play. The attached-line inequalities and give the displayed uniform control for every deleted tube. The next cubic scale is large enough for the induction, and the residue-class intersection is exactly the original liminf target rather than merely a subset.
Lemma 2.3No issue found in the reviewed proof
The open-submanifold patching lemma is valid as stated in v2
PDF p. 8 · Lemma 2.3 · arXiv:2608.24401v2
The lemma assumes that is closed, so is an open manifold on which the restricted game is defined. Intersection with the HAW complement and upward closure give one implication, while restriction gives the converse. The exceptional zero sets used in §7 are explicitly closed, so every application satisfies the lemma.
Lemmas 7.1–7.5No issue found in the reviewed proof
The ambient-to-curve transfer closes with the stated constants
PDF pp. 25–29 · §7 · arXiv:2608.24401v2
When the derivative of is large, the mean-value theorem confines a tube pullback to an interval of radius . If it is small, regularity forces , and the structured coefficient estimate makes every tube point leave . Shadow balls are nested and legal; choosing gives the one-dimensional budget. Transference identifies with the pullback of .
Proof of Theorem 1.8 and Appendix ANo issue found in the reviewed proof
The line estimate and both active examples are complete
PDF pp. 29–35 · proof of Theorem 1.8 and Appendix A · arXiv:2608.24401v2
For rational , the convergent denominators make both coordinate errors , so their product times every intermediate cutoff tends to zero. The Sierpiński-gasket example combines projection rigidity, Furstenberg microsets, homogeneity, Frostman restriction, and diffuse-set winning in a consistent order. The second example is the image of a product HAW set under a diffeomorphism and is disjoint from the chosen graph. An extra construction present only inside an inactive source block is excluded from the public v2 audit and graph.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.
04Sources7 reported findingsFurther source verification needed
The direct game-theoretic, Diophantine, dynamical, curve, line, and active fractal inputs were checked in exact versions or explicit public proof-bearing fallbacks and match their uses. The recursive graph nevertheless has nine disclosed original-proof access limits: five inherited from the ordinary Littlewood dimension branch and four in the active Sierpiński projection branch. Exact downstream statements and applicability were checked, but they do not replace inspection of those original proofs.
Direct game, Diophantine, curve, and line inputsApplicable and sufficient
The selected source theorems apply with the printed hypotheses
Focal PDF pp. 2–3, 7–9, 11–14, and 25–29 · arXiv:2608.24401v2
Broderick–Fishman–Kleinbock–Reich–Weiss and Fishman–Simmons–Urbański supply the HAW closure and potential-game facts. An and Badziahin–Pollington–Velani supply the attached-line and dual transference inputs; Nesharim–Simmons includes endpoint weights. An–Beresnevich–Velani Theorems 2.1 and 1.2 give the curve and line winning results, and Badziahin–Harrap–Nesharim–Simmons gives the one-dimensional absolute-winning upgrade. Their hypotheses match the focal uses.
Littlewood and homogeneous-dynamics inputsApplicable and sufficient
The exact downstream theorems give the claimed translations
Focal PDF pp. 2–5 and 22–23 · Corollaries 1.3–1.4, Lemma 6.1, and §1.4 · arXiv:2608.24401v2
Einsiedler–Katok–Lindenstrauss supplies the zero Hausdorff dimension of ordinary Littlewood exceptions and the orbit formulation. Bandi–Fregoli–Kleinbock gives the uniform multiplicative/fiberwise-divergence correspondence, while Kleinbock gives the endpoint-inclusive weighted bounded-orbit correspondence. Their coordinate conventions agree with the focal chart up to harmless permutations.
Margulis–Tomanov inputNot independently verified exact source proof
The original measure-rigidity proof remains unverified
Recursive source branch for Einsiedler–Katok–Lindenstrauss Theorem 1.5 · arXiv:2608.24401v2
The exact downstream article restates the Margulis–Tomanov results and checks their hypotheses, supporting the focal invocation. A proof-bearing copy of the original Margulis–Tomanov article was not accessible, so its original proof was not independently verified.
Margulis and Tomanov, Journal d'Analyse Mathématique 69 (1996), 25–54 ↗Prasad–Raghunathan and Ratner inputsNot independently verified exact source proofs
Two original proofs in the recursive compactness branch remain unverified
Recursive branch through Einsiedler–Katok–Lindenstrauss and Lindenstrauss–Weiss · arXiv:2608.24401v2
The exact downstream articles identify the compact-support and orbit-closure statements and apply them with compatible hypotheses. The proof-bearing Prasad–Raghunathan 1972 and Ratner 1991 articles remained behind official access gates; their two original proofs were not inspected.
Prasad and Raghunathan, Annals of Mathematics 96 (1972), 296–317 ↗Dani and Moore inputsNot independently verified exact source proofs
Two older proofs in the recursive Mautner branch remain unverified
Recursive source branch through Shah 1991 and Moore 1980 · arXiv:2608.24401v2
Exact downstream versions identify and apply the required Dani 1977 and Moore 1966 statements. The original proof-bearing articles were not accessible through their official records. No statement or hypothesis mismatch was found, but those two original proofs remain unverified.
Moore, American Journal of Mathematics 88 (1966), 154–178 ↗Sierpiński projection branchNot independently verified all exact source proofs
Four recursive originals behind projection rigidity remain access-limited
Focal PDF pp. 30–33 · Appendix A, Example 1 · chain through Shmerkin Corollary 6.3 and Hochman Theorem 1.6 · arXiv:2608.24401v2
The exact Shmerkin and Hochman Annals articles, the Käenmäki–Ojala–Rossi arXiv fallback, the Käenmäki–Rossi version of record, and the cited terminal books were inspected at the selected claims. Full recursion stops at Bourgain's implicit discretized structural theorem, Lau–Ngai Proposition 3.2, Kingman's subadditive theorem, and Esseen's Berry–Esseen estimate. Their roles are explicit downstream, but the four original proofs were not all available.
Shmerkin, Annals of Mathematics 189 (2019), 319–391 ↗Terminal published booksApplicable and sufficient
The remaining recursive branches end at identified standard monographs
Dependency graph terminal layers · arXiv:2608.24401v2
The graph stops transparently at published books for König's lemma, Frostman's lemma, additive combinatorics, differentiation and conditional-expectation limits, ergodic limits, recurrence, and standard homogeneous-dynamics background. They are marked as terminal sources rather than presented as freshly re-audited original proofs.