Waring's problem with restricted digits

TL;DR

This paper extends Waring's problem to integers with digits restricted to two coprime values in a given base, showing that large numbers can be expressed as sums of a bounded number of k-th powers from this set.

Contribution

It introduces a novel approach to Waring's problem with digit restrictions, establishing explicit bounds for representing large integers as sums of k-th powers.

Findings

Large integers are representable as sums of bounded k-th powers from restricted digit sets.

Provides explicit bounds on the number of summands needed.

Extends classical Waring's problem to digit-restricted contexts.

Abstract

Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then every sufficiently large integer is a sum of at most $b^{160 k^2}$ numbers of the form $x^k$, $x \in \mathcal{S}$.