Waring and Waring-Goldbach subbases with prescribed representation function

TL;DR

The paper develops a unified framework for constructing additive subbases with prescribed representation counts, using regular variation and probabilistic methods to control random subsets of $\mathbb{N}^k$ and $\mathbb{P}^k$. It proves that, for suitable $h$ and growth functions $F$, there exist $A$ with $r_{A,h}(n)$ asymptotic to a Waring–type singular series times $F(n)$, and, in Waring–Goldbach settings, to a singular series times a power $n^{\kappa}$ or to $n^{\kappa}\psi(n)$ under mild regularity. The results extend prior works by Vu, Wooley, and Pliego, providing sharp asymptotics and thin subbases, including congruence-respecting cases and prime-power constructions. Methodologically, the paper combines a probabilistic random-subset approach with concentration inequalities for boolean polynomials and a detailed circle-method treatment of prime powers, yielding both average-case and almost-sure construction results with explicit growth regimes. These constructions advance the understanding of how flexible growth conditions can be imposed on representation functions in Waring-type problems and open pathways for tailored subbases in additive number theory.

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 \[ r_{A,h}(n) \sim \mathfrak{S}^*_{k,h}(n) c n^κ\] for $0 < κ< h/k - 1$ and $c > 0$, where $\mathfrak{S}^*_{k,h}(n)$ is the singular series from Waring--Goldbach's problem. Additionally, for $0 \leq κ\leq h/k - 1$ and functions $ψ$ satisfying $ψ(x) \asymp_λ ψ(x^λ)$ for every $λ>0$, if $\log x \ll x^κ ψ(x) \ll x^{h/k-1}/(\log x)^h$, then there exists $A \subseteq \mathbb{P}^k$ with $r_{A,h}(n) \asymp n^κψ(n)$ for $n$ satisfying certain congruence conditions, producing thin subbases of prime powers when $κ= 0$, $ψ= \log$.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof