Pure Math Department Colloquium

Anand Pillay, University of Notre Dame

Quasirandomness of definable subsets of algebraic groups over finite fields

We give an arithmetic version of Tao’s algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs (G,D) where G is a group and D an arbitrary subset. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph (G,G,E) where E(x,y) is xy−1 ∈ D, witnessed by a the decomposition of G into cosets of a (uniformly definable) small index normal subgroup H of G. We compare to results of Green and Gowers. (This is joint work with Atticus Stonestrom.)

MC 5501