The celebrated Skolem-Mahler-Lech theorem asserts that if u := (un)n ≥ 0
is a linearly recurrent sequence of integers,
then the set of its zeros, that is the set of positive integers n such un = 0,
form a union of finitely many infinite arithmetic
progressions together with a (possibly empty) finite set. Except for some special cases, is not known how to bound
effectively all the zeros of u. This is called the Skolem problem.
In this talk we present the notion of a universal Skolem
Skolem set, which is an infinite set of positive integers S such that for every linearly
recurrent sequence u, the solutions
un = 0 with n ∈ S are effectively computable. We present
a couple of examples of universal Skolem sets, one of which
has positive lower density as a subset of all the positive integers.
It is joint a work with J. Ouaknine (Max Planck Saarbrücken), J. B. Worrell (Oxford).