Efficient congruencing in ellipsephic sets: the quadratic case

TL;DR

This paper develops bounds on solutions to quadratic Vinogradov systems within digit-restricted sets, leveraging digit set properties to improve over traditional bounds in additive number theory.

Contribution

It introduces a novel approach combining digital restrictions with quadratic Vinogradov systems to achieve sharper bounds, advancing the understanding of additive structures in restricted digit sets.

Findings

Improved bounds on solutions to quadratic Vinogradov systems with digit restrictions

Identification of digit sets that minimize representations of natural numbers

Enhanced methods for analyzing additive problems in digital sets

Abstract

In this paper, we bound the number of solutions to a quadratic Vinogradov system of equations in which the variables are required to satisfy digital restrictions in a given base. Certain sets of permitted digits, namely those giving rise to few representations of natural numbers as sums of elements of the digit set, allow us to obtain better bounds than would be possible using the size of the set alone.