By  Vitaly Bergelsohn  
From  Department of Mathematics The Ohio State University, Columbus, OH 

When  March 23, 2006 11:00 am 

Where  Engineering E206  
Abstract  Many familiar theorems in various areas of mathematics have the following common feature: if A is a large set, then the set of its differences, AA, is VERY large. For example:
(i) If A is a set of reals having positive Lebesgue measure, then there exists a positive real a, so that AA contains the interval (a,a). (ii) If A is a set of natural numbers having positive upper density, then for any polynomial p(n) having integer coefficients and zero constant term, the set AA contains infinitely many integers of the form p(n). (iii) If F is an infinite algebraic field and G is a subgroup of finite index in the multiplicative group F*, then GG = F. In this talk we shall discuss these and other similar results from the perspective of Ergodic Ramsey Theory. This discussion will lead us to new interesting results and conjectures. In particular we will see the foregoing as a special case of the appearance of rather arbitrary finite configurations inside sufficiently large sets. The talk is intended for a general audience. 

Further Information  Daniel Rudolph 