Isaac Goldbring : Progress on a sumset conjecture of Erdös

Posted by Isaac Goldbring , part of the Logic Seminar.

At

Aug. 27, 2013, 4 p.m.

In

SEO 427

Abstract

Erdös conjectured that any set A of natural numbers of <i>positive lower density</i> necessarily contains a sumset B+C, where B and C are infinite sets of natural numbers. In this talk, I will show how techniques from nonstandard analysis can be used to make progress on this conjecture. In particular, we settle this conjecture when A has <i> Banach density</i> exceeding 1/2 and use this result to prove a "one-translate" version of the conjecture for arbitrary A of positive Banach density. All necessary notions from combinatorial number theory and nonstandard analysis will be introduced. This work is joint with Mauro DiNasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg and was partially done during our Squares Week at AIM.