ERGODIC THEORY AND THE PROPERTIES OF LARGE SETS

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, A-A,
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 A-A 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 A-A 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 G-G = 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.