On certain bilinear sums with modular square roots and applications

TL;DR

This paper improves bounds on bilinear exponential sums involving modular square roots, extending previous work on additive energies, and advances understanding of the large sieve for square moduli.

Contribution

It extends bounds on additive energies of modular square roots and applies these to improve bounds on bilinear exponential sums and the large sieve for square moduli.

Findings

Enhanced bounds on bilinear exponential sums with modular square roots.

Partial progress on the large sieve inequality for square moduli.

Abstract

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.