When is $A + x A =\mathbb{R}$
Jinhe Ye, Liang Yu, Xuanheng zhao
TL;DR
The paper studies when additive subgroups, subrings, and subfields of $\mathbb{R}$ can satisfy $A+xA=\mathbb{R}$, using Hausdorff dimension as a size measure. It constructs a Borel $F_σ$ additive subgroup $A$ with $\dim_H(A)=\tfrac{1}{2}$ and some $x$ such that $A+xA=\mathbb{R}$, and, under CH, a dimension-zero subgroup with the property for all irrational $x$. A unifying thread blends geometric measure theory, descriptive set theory, and recursion-theoretic randomness to reveal a sharp contrast between subrings (which force $A=\mathbb{R}$ when $A+xA=\mathbb{R}$) and exotic CH-based constructions that yield highly fractal, yet rich, examples; analogous $p$-adic versions are discussed. The work also raises several open problems about CH-independence, the existence of Borel/analytic examples, and maximal subfields with prescribed dimensional and measurability properties, suggesting potential applications to other exotic real-sets constructions.
Abstract
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.