发表机构
Brandeis University; University of Chicago; Shanghai Institute for Mathematics and Interdisciplinary Sciences; Fudan University（布兰迪斯大学; 芝加哥大学; 上海数学与交叉学科研究院; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了BFK25提出的一致Littlewood猜想反例集合是超平面绝对获胜的，且在二维实数空间中具有满Hausdorff维数。
AI 中文摘要
本文证明了由\uc17BFK25提出的一致Littlewood猜想反例集合，即满足\ud835\udc44limsup\ud835\udc51→+∞ \ud835\udc51·min\ub9c5≤\ud835\udc51⟨\ud835\udc51\ud835\udc58⟩⟨\ud835\udc51\ud835\udc59⟩>0的实数对(\ud835\udc58,\ud835\udc59)构成的集合是超平面绝对获胜的，特别地，它在\ud835\udc52²中具有满Hausdorff维数。
英文摘要
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 $$ \limsup_{Q\to +\infty}\ Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs $(x,y)$ satisfying $$ \liminf_{m\to +\infty}\ Q_m\cdot \min_{1\leq q\leq Q_m}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning if $Q_{m+1} \gg Q_m^τ$ for some $τ> 1$. In particular, the above sets have full Hausdorff dimension in $\mathbb{R}^2$. 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.
Comments35 pages
