Universal Skolem Sets

Abstract

It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences, namely whether a given such sequence has a zero term. In this paper we introduce the notion of a Universal Skolem Set: an infinite subset $\mathcal{S}$ of the positive integers such that there is an effective procedure that inputs a linear recurrence sequence u = (u(n))n ≥ 0 and decides whether u(n) = 0 for some $n \in \mathcal{S}$. The main technical contribution of the paper is to exhibit such a set.