Waring and Waring-Goldbach subbases with prescribed representation function (2501.08371v2)

Abstract: We study $r_{A,h}(n)$, the number of representations of integers $n$ as sums of $h$ elements from subsets $A$ of $k$-th powers $\mathbb{N}^k$ and $k$-th powers of primes $\mathbb{P}^k$ for $k \geq 1$. Extending work by Vu, Wooley, and others, we show that for $h \geq h_k = O(8^k k^2)$, if $F$ is a regularly varying function satisfying $\lim_{n\to\infty} F(n)/\log n = \infty$, then 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 from Waring's problem. For $h \geq 2k^2(2\log k + \log\log k + O(1))$, we show the existence of $A \subseteq \mathbb{P}^k$ with [ r_{A,h}(n) \sim \mathfrak{S}^*_{k,h}(n) c n^\kappa ] for $0 < \kappa < h/k - 1$ and $c > 0$, where $\mathfrak{S}^*_{k,h}(n)$ is the singular series from Waring--Goldbach's problem. Additionally, for $0 \leq \kappa \leq h/k - 1$ and functions $\psi$ satisfying $\psi(x) \asymp_{\lambda} \psi(x^{\lambda})$ for every $\lambda>0$, if $\log x \ll x^{\kappa} \psi(x) \ll x^{h/k-1}/(\log x)^h$, then there exists $A \subseteq \mathbb{P}^k$ with $r_{A,h}(n) \asymp n^\kappa \psi(n)$ for $n$ satisfying certain congruence conditions, producing thin subbases of prime powers when $\kappa = 0$, $\psi = \log$.