We solve the mod G Cauchy functional equation f(x+y) = f(x) + f(y) (mod G), where G is a countable subgroup of ℝ and f:ℝ → ℝ is Borel measurable. We show that the only solutions are functions linear mod G.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-9,
author = {Nikos Frantzikinakis},
title = {Additive functions modulo a countable subgroup of $\mathbb{R}$},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {117-122},
zbl = {1016.39018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-9}
}
Nikos Frantzikinakis. Additive functions modulo a countable subgroup of ℝ. Colloquium Mathematicae, Tome 96 (2003) pp. 117-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-9/