ENERGY BOUNDS FOR MODULAR ROOTS AND THEIR APPLICATIONS

Bryce Kerr; Ilya D. Shkredov; Igor E. Shparlinski; Alexandru Zaharescu

doi:10.1017/S1474748023000397

ENERGY BOUNDS FOR MODULAR ROOTS AND THEIR APPLICATIONS

Bryce Kerr
, Ilya D. Shkredov
, Igor E. Shparlinski
, Alexandru Zaharescu

Mathematics

Research output: Contribution to journal › Article › peer-review

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 Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

Original language	English (US)
Pages (from-to)	1765-1806
Number of pages	42
Journal	Journal of the Institute of Mathematics of Jussieu
Volume	24
Issue number	5
Early online date	Mar 4 2024
DOIs	https://doi.org/10.1017/S1474748023000397
State	Published - Sep 1 2025

Keywords

Modular roots
arithmetic combinatorics
geometry of numbers

ASJC Scopus subject areas

General Mathematics

Online availability

10.1017/S1474748023000397

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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 Sali{\'e} sums and to a new equidistribution estimate for the set of modular roots of primes.",

keywords = "Modular roots, arithmetic combinatorics, geometry of numbers",

author = "Bryce Kerr and Shkredov, \{Ilya D.\} and Shparlinski, \{Igor E.\} and Alexandru Zaharescu",

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AU - Shparlinski, Igor E.

AU - Zaharescu, Alexandru

N1 - During the preparation of this work, B.K. was supported by the Academy of Finland Grant 319180 and by the Max Planck Institute for Mathematics, I.D.S. by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2023-934), and I.E.S. by the Australian Research Council Grants DP170100786 and DP200100355.

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Y1 - 2025/9/1

N2 - 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 Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

AB - 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 Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

KW - Modular roots

KW - arithmetic combinatorics

KW - geometry of numbers

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ENERGY BOUNDS FOR MODULAR ROOTS AND THEIR APPLICATIONS

Abstract

Keywords

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