Gowers uniformity of thin subsets of primes - Institut Élie Cartan de Lorraine

Date/heure
3 mars 2022
14:30 - 15:30

Lieu
Salle de séminaire de Théorie des Nombres virtuelle

Oratrice ou orateur
Fernando Xuancheng Shao (University of Kentucky)

Catégorie d'évènement
Séminaire de Théorie des Nombres de Nancy-Metz

Résumé

A celebrated theorem of Green-Tao asserts that the set of primes contains arbitrarily long arithmetic progressions. In fact, they count asymptotically the number of $k$ -term arithmetic progressions in primes up to a threshold. Their work involves discorrelation estimates between primes and nilsequences, which imply that the set of primes is Gowers uniform. In this talk I will discuss results of this type for primes restricted to short intervals and in arithmetic progressions. For example, we prove that the set of primes in $(X, X+H]$ with $H > X^{5/8+\varepsilon}$ is Gowers uniform; we also prove that, for almost all $q < X^{1/4-\varepsilon}$ , the set of primes up to $X$ in a coprime residue class $a\pmod{q}$ is Gowers uniform. This is based on joint works with K. Matomäki, J. Teräväinen, T. Tao.