Dirichlet sets and Erdos-Kunen-Mauldin theorem

Details Dirichlet sets and Erdos-Kunen-Mauldin theorem Dirichlet sets and Erdos-Kunen-Mauldin theorem

2007-12-13

arxiv.org/abs/0712.2112v1

Mathematics

General Topology

9 pages

Scientific paper

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.

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