On Vu's theorem in Waring's problem for thinner sequences

TL;DR

This paper extends Vu's theorem in Waring's problem to thinner sequences, establishing asymptotic formulas for the number of representations of integers as sums of k-th powers within specific subsequences, and improves understanding of minimal representations.

Contribution

It introduces new subsequences of k-th powers for which Waring's problem asymptotics hold, and refines bounds on the number of representations for large integers, addressing questions posed by Vu and Wooley.

Findings

Existence of subsequences with asymptotic representation formulas for almost all integers.

For s ≥ T(k), sequences exist with representation counts comparable to log n.

Results sharpen previous bounds and address open questions in Waring's problem.

Abstract

Let $k\in \mathbb{N}$ and $s\geq k(\log k+3.20032)$. Let $\mathbb{N}_{0}^{k}$ be the set of $k$-th powers of nonnegative integers. Assume that $\psi$ is an increasing function tending to infinity with $\psi(x)=o(\log x)$ and satifying some regularity conditions. Then, there exists a subsequence $\mathfrak{X}_{k}=\mathfrak{X}_{k}(s)\subset\mathbb{N}_{0}^{k}$ for which the number of representations $R_{s}(n;\mathfrak{X}_{k})$ of each $n\in\mathbb{N}$ as $$n=x_{1}^{k}+\ldots+x_{s}^{k}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_{i}^{k}\in\mathfrak{X}_{k}$$ satisfies the asymptotic formula $$ R_{s}(n;\mathfrak{X}_{k})\sim \mathfrak{S}(n)\psi(n)$$ for almost all natural numbers $n$, with $\mathfrak{S}(n)$ being the singular series associated to Waring's problem. If moreover $s\geq k(\log k+4.20032)$ the above conclusion holds for almost all $n\in [X,X+\log X]$ as $X\to\infty$. Let $T(k)$ be the least natural number for which it is known that all large integers are the sum of $T(k)$ $k$-th powers of natural numbers. We also show for $k\geq 14$ and every $s\geq T(k)$ the existence of a sequence $\mathfrak{X}_{k}'\subset \mathbb{N}_{0}^{k}$ satisfying $$R_{s}(n;\mathfrak{X}_{k}')\asymp \log n$$ for every sufficiently large $n$. The latter conclusion sharpens a result of Wooley and addresses a question of Vu.