Effective Bounds on Dependence among Iterates of a Rational Function modulo S-Integers

Abstract: 

A famous result of Northcott's states that the set of preperiodic points of a rational function in a number field has finite size. Recently there has been interest in the arithmetic structure of orbits. Berczes, Ostafe, Shparlinski and Silverman (2019) extended Northcott's Theorem to the case where points in an orbit need only be equal away from a fixed finite set of primes. Their proof relies crucially on a Diophantine approximation result obtained using Roth’s Theorem. We will discuss the background of this work and present an effective variant of their result based on a new upper bound on the S-parts of the values of rational functions over a number field.