Energy bounds for modular roots and their applications

TL;DR

This paper improves bounds on additive energies of modular roots using advanced mathematical techniques and applies these results to bounds on Salié sums and equidistribution of modular roots of primes.

Contribution

It generalizes and enhances recent bounds for additive energies of modular roots, integrating methods from additive combinatorics, algebraic number theory, and geometry of numbers.

Findings

Improved bounds on additive energies of modular roots

New bounds on correlations between Salié sums

A novel equidistribution estimate for modular roots of primes

Abstract

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between {\it Sali{\'e}} sums and to a new equidistribution estimate for the set of modular roots of primes.