The Dynamical Mordell-Lang problem

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational self-map map f on X, any subvariety Y of X, and any point x in X whose orbit under f is in the domain of definition for f, the set S is a finite union of arithmetic progressions together with a set of Banach density zero. We prove a similar result for the backward orbit of a point

Primitive geodesic lengths and (almost) arithmetic progressions

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma