HOẠT ĐỘNG TRONG TUẦN

Quadratic congruences and Weyl sums
Người báo cáo: Ngô Trung Hiếu

Thời gian: 14h30, Thứ bảy 11/12/2021

Tóm tắt: Given a sequence of real numbers arising from an arithmetic context, how are their fractional parts distributed on the unit interval? There are many beautiful instances of this question in number theory.

In a classical context, we may consider a quadratic congruence, varying the moduli and forming a sequence of roots-to-moduli ratios. In 1963, Christopher Hooley showed that if the moduli vary over the natural numbers, then the sequence is uniformly distributed modulo one. The equidistribution modulo one when the moduli take values in the prime numbers is a much more challenging problem. It was first established for negative-discriminant quadratics by Duke-Friedlander-Iwaniec in 1995 and then for positive-discriminant quadratics by Toth in 2012. In both cases, a crucial ingredient is a sufficiently strong estimate for exponential sums of Weyl type associated with the congruence roots.

We plan on giving a general introduction to this circle of equidistribution questions. We will review the classical results of Hooley, Duke-Friedlander-Iwaniec, and Toth. We will discuss our recent improvements on bounds of congruence Weyl sums for positive-discriminant quadratics. Time permitting, we will highlight interesting applications of congruence Weyl sums.

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