$δ$-Badly approximable numbers and ubiquitously losing sets
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| Format: | Preprint |
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2026
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| _version_ | 1866910220791316480 |
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| author | Tseng, Jimmy |
| author_facet | Tseng, Jimmy |
| contents | We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(δ) \subset \boldsymbol{\operatorname{Bad}}(δ')$ for $δ\geq δ'>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $τ$-well approximable numbers. We show that the set $\boldsymbol{\operatorname{Bad}}(δ)$ is a $(1/3, 18 δ)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(α, β)$-$\textit{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(α, β)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\boldsymbol{\operatorname{Bad}}(δ)$ is a $(1/2, 18/δ)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\boldsymbol{\operatorname{Bad}}(δ)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\boldsymbol{\operatorname{Bad}}(δ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06325 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $δ$-Badly approximable numbers and ubiquitously losing sets Tseng, Jimmy Number Theory Dynamical Systems 11J83, 28A80, 11K55, 91A05 We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(δ) \subset \boldsymbol{\operatorname{Bad}}(δ')$ for $δ\geq δ'>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $τ$-well approximable numbers. We show that the set $\boldsymbol{\operatorname{Bad}}(δ)$ is a $(1/3, 18 δ)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(α, β)$-$\textit{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(α, β)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\boldsymbol{\operatorname{Bad}}(δ)$ is a $(1/2, 18/δ)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\boldsymbol{\operatorname{Bad}}(δ)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\boldsymbol{\operatorname{Bad}}(δ)$. |
| title | $δ$-Badly approximable numbers and ubiquitously losing sets |
| topic | Number Theory Dynamical Systems 11J83, 28A80, 11K55, 91A05 |
| url | https://arxiv.org/abs/2605.06325 |