Generalized Kempner Series and the Erdos Conjecture on Arithmetic Progressions

By Alex Walker

September 18, 2012

Abstract

The Erdos Conjecture on Arithmetic Progressions proposes that integer sets with divergent harmonic sums contain arithmetic progressions of arbitrary (finite) length. Equivalently, one may ask whether the harmonic sums of integer sets without arithmetic progressions of length n ('n-free' sets) are bounded. In this talk, we'll show that a class of sets with fractal structure is n-free for some (effective) n, and that sets of this form can have large harmonic sums. As time permits, we'll discuss some ongoing work concerning estimates on a few arithmetic functions which arise during the previous exposition.