Waring--Goldbach subbases with prescribed representation functions

Abstract: We investigate representation functions $r_{A,h}(n)$ of subsets $A$ of \( k \)-th powers \( \mathbb{N}^k \) and \( k \)-th powers of primes \( \mathbb{P}^k \). Building on work of Vu, Wooley, and others, we prove that for \( h \geq h_k = O(8^k k^2) \) and regularly varying \( F(n) \) satisfying \( \lim_{n\to\infty} F(n)/\log n = \infty \), there exists \( A \subseteq \mathbb{N}^k \) such that \[ r_{A,h}(n) \sim \mathfrak{S}_{k,h}(n) F(n), \] where $\mathfrak{S}_{k,h}(n)$ is the singular series associated to Waring's problem. In the case of prime powers, we obtain analogous results for \( F(n) = n^{\kappa} \). For \( F(n) = \log n \), we prove that for every \( h \geq 2k^2(2\log k + \log\log k + O(1)) \), there exists \( A \subseteq \mathbb{P}^k \) such that \( r_{A,h}(n) \asymp \log n \), showing the existence of thin subbases of prime powers.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)