Universal Skolem Sets

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 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 ∈ S . The main technical contribution of the paper is to exhibit such a set.