## On a theorem of Hildebrand.(English)Zbl 1450.11017

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.

### MSC:

 11B75 Other combinatorial number theory

### Keywords:

IP-set; multiplicative subgroup

Zbl 0727.11036
Full Text:

### 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
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