JUXTAPOSING COMBINATORIAL AND ERGODIC PROPERTIES OF LARGE SETS OF INTEGERS

The classical approach of Furstenberg allows one to associate with any large set E subset of Z a dynamical system X-E = (X, B, mu, T) which "encodes" the combinatorial properties of E via the multiple recurrence properties of the transformation T. While one can always assume without loss of generality that X-E is ergodic, the requirement of ergodicity of T-2 puts rather stringent combinatorial constraints on the set E. We undertake a close study of the connection between the combinatorial richness of large sets in Z and ergodic properties of the corresponding system X-E. In particular, we characterize, in combinatorial terms, totally ergodic (resp. weakly mixing) sets E, i.e., sets for which T is totally ergodic (resp. weakly mixing). This leads to numerous new combinatorial applications.