Seminario de Teoría de Números

TITLE: A non-flag arithmetic regularity lemma and counting lemma

SPEAKER: Daniel Altman (University of Oxford

DATE & TIME: lunes 14 de noviembre de 2022 - 17:30

VENUE: Seminario online por Microsoft Teams. Enlace para acceder al aula virtual:

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ABSTRACT: Green and Tao's arithmetic regularity lemma allows one to decompose a bounded function on {1, 2, ..., N} (e.g. a characteristic function of a set) into an arithmetically-structured function (a nilsequence), a function that is small in L^2, and a function which is very arithmetically random (has very small Gowers norm). For certain problems in additive combinatorics, the latter two functions can be shown to contribute negligibly, and so the problem is reduced to an analysis of nilsequences. Green and Tao provide a complementary counting lemma which, via the equidistribution of polynomial sequences on nilmanifolds, can reduce the relatively difficult arithmetic problem of analysing nilsequences to a more algebraic problem on nilmanifolds. Green and Tao's counting lemma applies to linear patterns which satisfy a particular algebraic criterion known as the "flag condition".

In this talk we will flesh out all of the statements in the previous paragraph for a general audience. We will then describe motivating examples for, and the main ideas in, recent work which produces an arithmetic regularity lemma and counting lemma which together apply to all linear patterns.