On a theorem of Hildebrand

Abstract

We prove that for each multiplicative subgroup $A$ of finite index in ${\mathbb{Q}}^{+}$, the set of integers $a$ with $a,a+1\in A$ is an IP-set. This generalizes a theorem of Hildebrand concerning completely multiplicative functions taking values in the $k$-th roots of unity.

3 pages, Comments welcome