Dirichlet Sets, Erdős-kunen-mauldin Theorem, and Analytic Subgroups of the Reals

We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich’s theorem saying that every proper analytic subgroup of the real line is included in an Fσ set of measure zero. Using the strengthened theorems we generalize the fact that permitted sets for families of trigonometric thin sets are perfectly meager.