Work

On a family of sparse exponential sums

Year: 2024

Type: article

Abstract: We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro‐geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.

Source: Mathematische Nachrichten

DOI: 10.1002/mana.202300426

Authors: Moubariz Z. Garaev, Zeev Rudnick, Igor E. Shparlinski

Institutions: Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Tel Aviv University, UNSW Sydney

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,300

Funders: European Commission, Israel Science Foundation, European Research Council, Australian Research Council