When is $A + x A =\mathbb{R}$

TL;DR

The paper constructs special subgroups of reals with particular additive properties, exploring conditions under which their sums with scaled versions cover the entire real line, using techniques from recursion theory and randomness.

Contribution

It demonstrates the existence of additive subgroups with specific Hausdorff dimensions that satisfy certain sum conditions, introducing new methods and extending to p-adic contexts.

Findings

Existence of an additive $F_\sigma$ subgroup with Hausdorff dimension 1/2 satisfying $A + xA = eals$.

If $A$ is a subring and $A + xA = eals$, then $A$ must be all of $ eals$.

Under CH, existence of a subgroup with Hausdorff dimension 0 satisfying $A + xA = eals$ for all $x$.

Abstract

We show that there is an additive $F_\sigma$ 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.