Waring and Waring-Goldbach subbases with prescribed representation function

TL;DR

This paper develops a probabilistic principle for constructing subbases with prescribed growth of representation functions, applied to powers and prime powers, with results on their asymptotic behaviors.

Contribution

It introduces a general probabilistic subbasis principle allowing the realization of specific growth patterns for representation functions of subbases.

Findings

Existence of subbases with prescribed regularly varying growth functions.

Construction of thin subbases of prime powers with logarithmic representation functions.

Application to powers and prime powers demonstrating the flexibility of the method.

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.