Measure Zero Sets Whose Algebraic Sum Is Non-Measurable

Abstract

In this note we will show that for every natural number $n>0$ there exists an $S\subset[0,1]$ such that its $n$-th algebraic sum $nS=S+\cdots +S$ is a nowhere dense measure zero set, but its $n+1$-st algebraic sum $nS+S$ is neither measurable nor it has the Baire property. In addition, the set $S$ will be also a Hamel base, that is, a linear base of $\mathbb{R}$ over $\mathbb{Q}$.