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

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.