F -sets and finite automata
Jason P. Bell, Rahim Moosa · Journal de Théorie des Nombres de Bordeaux · 2019
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.