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Winning property of counterexamples to Uniform Littlewood's Conjecture
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 (x,y) satisfying limsupQ→ +∞ Q· min1≤ q≤ Q⟨ qx⟩⟨ qy⟩>0, is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs (x,y) satisfying liminfm→ +∞ Qm· min1≤ q≤ Qm⟨ qx⟩⟨ qy⟩>0, is hyperplane absolute winning if Qm+1 ≫ Qmτ for some τ> 1. In particular, the above sets have full Hausdorff dimension in ℝ2. In addition, we prove that these sets are absolute winning on every regular C2 planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.
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BibTeX
@misc{neckrasov2026winning,
title = {{Winning property of counterexamples to Uniform Littlewood's Conjecture}},
author = {Vasiliy Neckrasov and Chengyang Wu and Bohan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24401},
eprint = {2608.24401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24401},
}
RIS
TY - UNPB
TI - Winning property of counterexamples to Uniform Littlewood's Conjecture
AU - Vasiliy Neckrasov
AU - Chengyang Wu
AU - Bohan Yang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.24401
UR - https://arxiv.org/abs/2608.24401
AB - 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 (x,y) satisfying limsup_Q→ +∞ Q· min_1≤ q≤ Q⟨ qx⟩⟨ qy⟩>0, is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs (x,y) satisfying liminf_m→ +∞ Q_m· min_1≤ q≤ Q_m⟨ qx⟩⟨ qy⟩>0, is hyperplane absolute winning if Q_m+1 ≫ Q_m^τ for some τ> 1. In particular, the above sets have full Hausdorff dimension in ℝ^2. In addition, we prove that these sets are absolute winning on every regular C^2 planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.
ER -