Efficient congruencing in ellipsephic sets: the general case

TL;DR

This paper develops bounds on solutions to Vinogradov systems with digit restrictions, leveraging additive structure in sets like squares, leading to improved results in the context of ellipsephic sets.

Contribution

It introduces a novel approach to bounding solutions of Vinogradov systems with digital restrictions, especially for sets like squares, enhancing previous bounds with additive structure.

Findings

Better bounds for solutions using digit restrictions

Diagonal behavior achieved with 2k(k+1) variables for squares

Improved understanding of solutions in ellipsephic sets

Abstract

In this paper, we bound the number of solutions to a general Vinogradov system of equations $x_1^j+\dots+x_s^j=y_1^j+\dots+y_s^j$, $(1\leq j\leq k)$, as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set -- the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behaviour with $2k(k+1)$ variables.