Additive functions modulo a countable subgroup of R

Abstract

We solve the mod $G$ Cauchy functional equation $$ f(x+y)=f(x)+f(y)\pmod G, $$ where $G$ is a countable subgroup of ${\mathbb R}$ and $f:{\mathbb R}\to {\mathbb R}$ is Borel measurable. We show that the only solutions are functions linear mod $G$.

Locations

• Colloquium Mathematicum - View - PDF