  The Skolem Problem asks to determine whether a given integer linear
recurrence sequence has a zero term. This problem arises across a wide range of
topics in computer science, including loop termination, formal languages,
automata theory, and control theory, amongst many others. Decidability of the
Skolem Problem is notoriously open. The state of the art is a decision
procedure for recurrences of order at most 4: an advance achieved some 40 years
ago, based on Baker's theorem on linear forms in logarithms of algebraic
numbers.
  A new approach to the Skolem Problem was recently initiated via the notion of
a Universal Skolem Set: a set $S$ of positive integers such that it is
decidable whether a given non-degenerate linear recurrence sequence has a zero
in $S$. Clearly, proving decidability of the Skolem Problem is equivalent to
showing that $\mathbb{N}$ itself is a Universal Skolem Set. The main
contribution of the present paper is to construct a Universal Skolem Set that
has lower density at least $0.29$. We show moreover that this set has density
one subject to the Bateman-Horn conjecture. The latter is a central unifying
hypothesis concerning the frequency of prime numbers among the values of
systems of polynomials.
