Efficient congruencing in ellipsephic sets: the general case

Abstract

In this paper, we bound the number of solutions to a general Vinogradov system of equations [Formula: see text] 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 behavior with [Formula: see text] variables.

Character sums over sparse elements of finite fields

“…Recently, motivated by a solution to the Gelfond problem by Mauduit and Rivat [23], there has been an explosion in the investigation of arithmetic problems involving integers with various digits restrictions in a given integer base $$g$$. For example, Bourgain [2, 3] has investigated primes with prescribed digits on a positive proportion of positions in their digital expansion, while Maynard [24, 25] has investigated primes with missing digits, see also [4, 9, 10, 26, 31, 35] for a series of other results about primes and other arithmetically interesting integers such as smooth and squarefree numbers with restrictions on their digits; Mauduit and Rivat [22] and Maynard [25] have also studied values of integral polynomials with digital restrictions; Bounds of exponential sums with digitally restricted integers can be found in [1, 12, 20, 29, 30, 32, 36]; in some of these, also the Goldbach and Waring problems with such numbers has been considered; Éminyan [11] studied average values of arithmetic functions for such numbers. …”