Universal Skolem Sets..

Coauthors: J. Ouaknine (Max--Planck Saabr"ucken), J. B. Worrell (Oxford). The celebrated Skolem--Mahler--Lech theorem asserts that if ${\bf u}:=(u_n)_{n\ge 0}$ is a linearly recurrent sequence of integers then the set of its zeros, that is the set of positive integers $n$ such $u_n=0$, 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 ${\bf u}$. This is called {\it the Skolem problem}. In this talk we present the notion of a {\it universal Skolem set}, which an infinite set of positive integers ${\mathcal S}$ such that for every linearly recurrent sequence ${\bf u}$, the solutions $u_n=0$ with $n\in {\mathcal S}$ 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.