可良好与恶劣逼近集合及快速获胜策略
Well and badly approximable sets, and rapid winning
浏览论文内容
中文总结 AI 辅助
该研究引入Ψ-快速博弈确定τ-逼近数集与非齐次恶劣逼近数集交集的豪斯多夫维数，得到了τ≥1时该交集的精确Jarník–Besicovitch维数。
中文摘要 AI 辅助
τ-逼近数集𝒲(τ)具有真正的分数维豪斯多夫维数，而非齐次恶劣逼近数集Bad^γ则具有满豪斯多夫维数。我们通过引入Ψ-快速博弈来确定二者交集的豪斯多夫维数，该博弈是Hatefi与Simmons（2024年预印本）提出的快速博弈的尺度敏感细化。对每个逼近函数ψ，我们证明𝒲(ψ)∩Bad^γ对由ψ确定的自然规范Ψ而言是强Ψ-快速获胜的。与获胜性质总隐含满豪斯多夫维数的Schmidt型博弈不同，Ψ-快速博弈针对指定的丢番图尺度校准，因此所得维数界明确依赖于Ψ的衰减。特别地，当ψ(q)=q^(-τ)（τ≥1）时，我们得到精确的Jarník–Besicovitch维数，即HD(𝒲(τ)∩Bad^γ)=2/(τ+1)。
英文摘要
The set of $τ$-approximable numbers, $\mathcal W(τ)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^γ$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $Ψ$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $ψ$, we prove that $\mathcal W(ψ)\cap\Bad^γ$ is strong $Ψ$-rapid winning for a natural gauge $Ψ$ determined by $ψ$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $Ψ$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $Ψ$. In particular, for $ψ(q)=q^{-τ}, τ\ge1,$ we recover the exact Jarník--Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(τ)\cap\Bad^γ\bigr)=\frac{2}{τ+1}.$$
发表机构
- La Trobe University（拉筹伯大学）
机构由 AI 辅助整理，请以论文原文为准。
