When is $A + x A =\mathbb{R}$
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| Format: | Preprint |
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2025
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| _version_ | 1866915996567076864 |
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| author | Ye, Jinhe Yu, Liang zhao, Xuanheng |
| author_facet | Ye, Jinhe Yu, Liang zhao, Xuanheng |
| contents | We show that there is an additive $F_σ$ subgroup $A$ of $\mathbb{R}$ and $x \in \mathbb{R}$ such that $\mathrm{dim_H} (A) = \frac{1}{2}$ and $A + x A =\mathbb{R}$. However, if $A \subseteq \mathbb{R}$ is a subring of $\mathbb{R}$ and there is $x \in \mathbb{R}$ such that $A + x A =\mathbb{R}$, then $A =\mathbb{R}$. Moreover, assuming the continuum hypothesis (CH), there is a subgroup $A$ of $\mathbb{R}$ with $\mathrm{dim_H} (A) = 0$ such that $x \not\in \mathbb{Q}$ if and only if $A + x A =\mathbb{R}$ for all $x \in \mathbb{R}$. A key ingredient in the proof of this theorem consists of some techniques in recursion theory and algorithmic randomness. We believe it may lead to applications to other constructions of exotic sets of reals. Several other theorems on measurable, and especially Borel and analytic subgroups and subfields of the reals are presented. We also discuss some of these results in the $p$-adics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When is $A + x A =\mathbb{R}$ Ye, Jinhe Yu, Liang zhao, Xuanheng Logic Classical Analysis and ODEs Group Theory Number Theory 28A80, 28A05, 03D32, 12L99 We show that there is an additive $F_σ$ subgroup $A$ of $\mathbb{R}$ and $x \in \mathbb{R}$ such that $\mathrm{dim_H} (A) = \frac{1}{2}$ and $A + x A =\mathbb{R}$. However, if $A \subseteq \mathbb{R}$ is a subring of $\mathbb{R}$ and there is $x \in \mathbb{R}$ such that $A + x A =\mathbb{R}$, then $A =\mathbb{R}$. Moreover, assuming the continuum hypothesis (CH), there is a subgroup $A$ of $\mathbb{R}$ with $\mathrm{dim_H} (A) = 0$ such that $x \not\in \mathbb{Q}$ if and only if $A + x A =\mathbb{R}$ for all $x \in \mathbb{R}$. A key ingredient in the proof of this theorem consists of some techniques in recursion theory and algorithmic randomness. We believe it may lead to applications to other constructions of exotic sets of reals. Several other theorems on measurable, and especially Borel and analytic subgroups and subfields of the reals are presented. We also discuss some of these results in the $p$-adics. |
| title | When is $A + x A =\mathbb{R}$ |
| topic | Logic Classical Analysis and ODEs Group Theory Number Theory 28A80, 28A05, 03D32, 12L99 |
| url | https://arxiv.org/abs/2505.00556 |