F-sets and finite automata

Abstract

It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a k-automatic subset of the natural numbers to that of an F-automatic subset of a finitely generated abelian group Γ equipped with an endomorphism F. Applied to the Mordell–Lang context, where F is the Frobenius action on a commutative algebraic group G over a finite field, and Γ is a finitely generated F-invariant subgroup of G, it is shown that the “F-subsets” of Γ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X⊆G is a closed subvariety then X∩Γ 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∩Γ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen’s Skolem–Mahler–Lech theorem to the Mordell–Lang context.