Dirichlet sets and Erdos-Kunen-Mauldin theorem

TL;DR

This paper strengthens a theorem by Erdos, Kunen, and Mauldin by showing that for any perfect set, there exists a Dirichlet set of measure zero such that their sum covers the real line, and explores implications for additive sets and analytic subgroups.

Contribution

It proves a stronger version of the Erdos-Kunen-Mauldin theorem with Dirichlet sets and investigates the structure of additive sets and subgroups in the real line.

Findings

Existence of Dirichlet sets that, when added to perfect sets, cover the real line.

All additive sets in certain families are perfectly meager in a transitive sense.

Every proper analytic subgroup is contained in an F-sigma set with a meager null sum.

Abstract

By a theorem proved by Erdos, Kunen and Mauldin, for any nonempty perfect set $P$ on the real line there exists a perfect set $M$ of Lebesgue measure zero such that $P+M=\mathbb{R}$. We prove a stronger version of this theorem in which the obtained perfect set $M$ is a Dirichlet set. Using this result we show that for a wide range of familes of subsets of the reals, all additive sets are perfectly meager in transitive sense. We also prove that every proper analytic subgroup $G$ of the reals is contained in an F-sigma set $F$ such that $F+G$ is a meager null set.