On a theorem of Hildebrand

TL;DR

This paper proves that for certain multiplicative subgroups of positive rationals, the set of integers with consecutive integers in the subgroup forms an IP-set, generalizing a theorem related to multiplicative functions.

Contribution

It extends Hildebrand's theorem to broader classes of multiplicative subgroups, showing the IP-set property for integers with consecutive elements in these groups.

Findings

The set of integers with both a and a+1 in the subgroup is an IP-set.

Generalization of Hildebrand's theorem to subgroups of finite index in 1^+.

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.