On the asymmetric additive energy of polynomials

TL;DR

This paper establishes that integer-valued polynomials exhibit low asymmetric additive energy by analyzing the scarcity of integer points on specific 4-dimensional affine hypersurfaces, contributing to understanding polynomial additive properties.

Contribution

It introduces a general result linking the scarcity of integer points on certain hypersurfaces to the small asymmetric additive energy of integer-valued polynomials.

Findings

Integer points are scarce on certain 4D affine hypersurfaces.

Integer-valued polynomials have small asymmetric additive energy.

Provides a new approach to studying polynomial additive properties.

Abstract

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy.