一致Littlewood猜想在具有正Hausdorff维数的集合上不成立
The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension
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中文总结 AI 辅助
该数学研究证明一致Littlewood猜想不成立，给出两类反例集合的Hausdorff维数下界，与经典Littlewood猜想的反例维数结论形成对比。
中文摘要 AI 辅助
由Bandi、Fregoli和Kleinbock提出的一致Littlewood猜想（ULC），在两数情形下断言：对所有实数ξ、ζ，当Q→∞时，Q乘以1≤n≤Q范围内‖nξ‖与‖nζ‖的最小值的极限为0。该猜想已被证明对几乎所有对(ξ,ζ)成立，但Schleischitz近期推翻了其完整表述，指出反例集合包含一个稠密的Gδ集。我们证明，第一个坐标为 badly approximable数的反例对集合的Hausdorff维数至少为3/2；还进一步证明，存在ζ使得(ξ,ζ)为ULC反例的badly approximable数ξ的集合具有完全Hausdorff维数，这与经典Littlewood猜想形成对比，后者已知其可能反例的Hausdorff维数为0。
英文摘要
The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|nξ\|\,\|nζ\|=0 $$ for all real $ξ,ζ$. It is proven to hold for almost every pair $(ξ,ζ)$. Schleischitz, however, has recently disproved the full statement and showed that the set of counterexamples contains a dense $G_δ$ set. We prove that a set of counterexample pairs with the first coordinate being a badly approximable number has Hausdorff dimension at least $3/2$. We further show that the set of badly approximable numbers $ξ$ for which there exists $ζ$ such that $(ξ,ζ)$ is a counterexample to ULC has full Hausdorff dimension. This contrasts with the classical Littlewood conjecture, for which the set of possible counterexamples is known to have Hausdorff dimension $0$.
