RÉSUMÉ

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 (or, sometimes, the set of distances between its elements) 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.

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