Waring and Waring-Goldbach subbases with prescribed representation function

Abstract

Let $h\geq 2$. For $A\subseteq \mathbb{N}$ write \[ r_{A,h}(n) := \#\{(x_1,\ldots,x_h)\in A^h ~|~ x_1+\cdots+x_h=n\}. \] We prove a general probabilistic subbasis principle: assuming an asymptotic for a weighted $h$-fold representation sum over a basis $B$, there exist subbases $A\subseteq B$ whose representation function $r_{A,h}(n)$ has prescribed regularly varying growth. We apply this to $k$-th powers $\mathbb{N}^k$ and to $k$-th powers of primes $\mathbb{P}^k$. For $h \geq k^2-k+O(\sqrt{k})$, we show that every regularly varying function $F$ with $F(x)/\log x\to\infty$ in the admissible range is realized, with the expected singular series factor. In particular, there exists $A\subseteq \mathbb{N}^k$ such that \[ r_{A,h}(n)\sim \mathfrak{S}_{k,h}(n) F(n). \] Moreover, in the prime setting we obtain thin subbases $A\subseteq \mathbb{P}^k$ with $r_{A,h}(n)\asymp \log n$ for $n$ in the admissible congruence classes.