On a theorem of Hildebrand. (English) Zbl 1450.11017

Dietzel, Carsten

Mosc. J. Comb. Number Theory 8, No. 2, 189-191 (2019).

A. Hildebrand [Mich. Math. J. 38, No. 2, 241–253 (1991; Zbl 0727.11036)] proved the following theorem:
For any fixed \(k \in \mathbb{Z}^+\), if \(f: \mathbb{Z}^+ \rightarrow \mathbb{C}\) is a completely multiplicative function (i.e., \(f (mn)= f (m) f (n)\) for all \(m, n \in \mathbb{Z}^+ \)) taking its values in the \(k\)-th roots of unity, then the set of \(a\in \mathbb{Z}^+\) fulfilling \( f (a) + f (a+1)= 1\) is nonempty.
The paper under review proves the following stronger statement: Let \(H\leq\mathbb{Q}^+\) be a (multiplicative) subgroup such that \(\mathbb{Q}^+/H\) is cyclic of finite order. Let \(H^*=H\cap \mathbb{Z}^+\). Then \(H^*\) is nonempty, as it is a “large set”, called IP-set. A set is an IP-set, if it contains all finite sums of an infinite sequence of positive integers.

Reviewer: László A. Székely (Columbia)

MSC:

11B75

Other combinatorial number theory

Keywords:

IP-set; multiplicative subgroup

Citations:

Zbl 0727.11036

References:

[1]	10.2307/2034033 · Zbl 0128.26802
[2]	10.1307/mmj/1029004331 · Zbl 0727.11036
[3]	10.2307/2033781 · Zbl 0104.03704
[4]	10.2307/2003130 · Zbl 0105.26501
[5]	10.4153/CJM-1963-020-4 · Zbl 0106.26002