Uniform syndeticity in multiple recurrence

Abstract

The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers d, l >= 1 and any epsilon > 0, we prove the existence of delta > 0 and K >= 1 (dependent only on d, l, and epsilon) such that the following holds: Consider a solvable group Gamma of derived length l, a probability space (X, mu), and d pairwise commuting measure-preserving Gamma-actions T-1, & mldr;, T-d on (X, mu). Let E be a measurable set in X with mu(E) >= epsilon. Then, K many (left) translates of {gamma is an element of Gamma: mu (T-1(gamma-1 )(E)boolean AND T-2(gamma-1)degrees T-1(gamma-1 )(E) boolean AND center dot center dot center dot boolean AND T-d(gamma-1 )degrees T-d-1(gamma-1 )degrees center dot center dot center dot degrees T-1(gamma-1 )(E)) >= delta} cover Gamma. This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers d, l >= 1 and any epsilon>0, there are delta>0 and K >= 1 (dependent only on d, l, and epsilon) such that for all finite solvable groups G of derived length l and any subset E subset of G(d) with m(circle times d)(E) >= epsilon (where m is the uniform measure on G), we have that K-many (left) translates of {g is an element of G:m(circle times d)({(a(1), & mldr;, a(n)) is an element of G(d): (a(1), & mldr;, a(n)), (ga(1), a(2), & mldr;, a(n)), & mldr;, (ga(1), ga(2), & mldr;, ga(n)) is an element of E}) >= delta} cover G. The proof of our main result is a consequence of an ultralimit version of Austin's amenable ergodic Szemeredi theorem.