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

Abstract

Let $k\in \mathbb{N}$ and $s\ge k(\log k+3.20032)$. Let ${\mathbb{N}}_0^k$ be the set of $k$-th powers of nonnegative integers. Assume that $\ddot{I}ˆ$ is an increasing function tending to infinity with $\ddot{I}ˆ(x)=o(\log x)$ 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 $$n=x_1^k+\dots +x_s^k{x}_i^k\in {\mathfrak{X}}_k$$ satisfies the asymptotic formula $$R_s(n;{\mathfrak{X}}_k)\sim \mathfrak{S}(n)\ddot{I}ˆ(n)$$ for almost all natural numbers $n$, with $\mathfrak{S}(n)$ being the singular series associated to Waring's problem. If moreover $s\ge k(\log k+4.20032)$ 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 $k\ge 14$ and every $s\ge T(k)$ the existence of a sequence ${\mathfrak{X}}_k^{\prime }\subset {\mathbb{N}}_0^k$ satisfying $$R_s(n;{\mathfrak{X}}_k^{\prime })\asymp \log n$$ for every sufficiently large $n$. The latter conclusion sharpens a result of Wooley and addresses a question of Vu.

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