Additive functions modulo a countable subgroup of $\mathbb{R}$

Nikos Frantzikinakis¹

¹ 331 McAllister Building State College, PA 16801, U.S.A.

Colloquium Mathematicum, Tome 95 (2003) no. 1, pp. 117-122.

Résumé

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 $G$.

Détail

DOI : 10.4064/cm95-1-9

Mots-clés : solve mod cauchy functional equation pmod where countable subgroup mathbb mathbb mathbb borel measurable only solutions functions linear mod