Apparaît dans la collection : 2018 - T1 - WS1 - Model theory and combinatorics

In this talk we present a stable version of the arithmetic regularity lemma for vector spaces over fields of prime order. The arithmetic regularity lemma for F n p (first proved by Green in 2005) states that given A ⊆ F n p , there exists H ≤ F n p of bounded index such that A is Fourier-uniform with respect to almost all cosets of H. In general, the growth of the index of H is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. Our main result is that, under a natural stability theoretic assumption, the bad bounds and non-uniform elements are not necessary. Specifically, we present an arithmetic regularity lemma for k-stable sets A ⊆ F n p , where the bound on the index of the subspace is only polynomial in the degree of uniformity, and where there are no non-uniform elements. This result is a natural extension to the arithmetic setting of the work on stable graph regularity lemmas initiated by Malliaris and Shelah. This is joint work with Julia Wolf.