Waring’s problem with digital restrictions

The aim of this paper is to consider an analogue of Waring’s problem with digital restrictions. In particular, we prove the following result. Letsq(n) be theq-adic sum of digits function and leth,m be fixed positive integers. Then fors>2k there existsn0∈ℕ such that each integern≥n0 has a representation of the form We will even give an asymptotic formula for the number of representations ofn in this way. The result is shown with help of the circle method in combination with a “digital” version of Weyl’s Lemma.