# Arithmetic statistics, discrete restriction, and Fourier analysis

February 15 to February 19, 2021

at the

American Institute of Mathematics, San Jose, California

organized by

Theresa Anderson, Frank Thorne, and Trevor Wooley

## Original Announcement

This workshop aims to explore several problems at the interface of harmonic analysis and analytic number theory, with an eye to bringing both groups of researchers together to make progress in discrete restriction, arithmetic statistics, exponential sum estimates and discrete harmonic analysis by using tools from both fields.

Number theory and analysis share many interactions, and there are several emerging areas where input from both fields will likely be quite fruitful. Arithmetic statistics is a subject focused on counting of objects of algebraic interest, has been extensively investigated by Bhargava and collaborators, and seems ripe for Fourier analytic input. Discrete restriction, as pioneered by Bourgain, is rooted in analysis but is sometimes amenable to number theoretic exponential sum estimates inaccessible to such tools as decoupling methods. Discrete analogues in harmonic analysis have been classified in many ways, but are frequently impeded by limited progress on deep number theoretic problems. By bringing together researchers from both analysis and number theory and having them interact on a variety of problems of emerging interest, we hope to make progress on several areas including:

• arithmetic statistics, including the use of Fourier analysis to improve key bounds
• discrete restriction for the curve $(x,x^3)$ and related problems
• discrete operators in harmonic analysis, such as the spherical maximal function in dimension 4
• exponential sum estimates arising from problems at the interface of analysis and number theory - connections between these areas

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Hilbert transforms and the equidistribution of zeros of polynomials
by  Emanuel Carneiro, Mithun Kumar Das, Alexandra Florea, Angel V. Kumchev, Amita Malik, Micah B. Milinovich, Caroline Turnage-Butterbaugh, and Jiuya Wang