F-sets and finite automata

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $\Gamma$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $\Gamma$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $\Gamma$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subvariety then X intersect $\Gamma$ is F-automatic. Derksen's notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X intersect $\Gamma$ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by Christopher Hawthorne; it is no longer claimed that an F-normal subset has a finite symmetric difference with an F-subset. The main theorems of the paper remain unchange

The Positivity Set of a Recurrence Sequence

We consider real sequences $(f_n)$ that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both for the special case just mentioned and in general