Efficient congruencing in ellipsephic sets: the quadratic case

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. In particular, when the digits are required to be squares, we obtain diagonal behaviour with 12 variables.