On Vu's theorem in Waring's problem for thinner sequences
Javier Pliego
TL;DR
This work advances Vu’s theorem in Waring’s problem by constructing thinner subsequences of k-th powers for which the s-fold representation counts have controlled asymptotics. It blends the circle method with weighted smooth Weyl sums (via Bru–Woo technology) and a probabilistic thinning framework to obtain both asymptotic formulas and concentration results for R_s(n;X_k) and R_s(n;X_k'). It delivers (i) an asymptotic R_s(n;X_k) ∼ 𝔖(n)ψ(n) for almost all n when s ≥ k(log k+4.20032) and short-interval analogues, and (ii) thinner-sequence constructions yielding R_s(n) ≍ log n for large n (k≥14) and concentration-based upper bounds beyond the logarithmic barrier. The results address questions of Vu and Wooley, sharpen limits in Waring-type problems for thinner sequences, and provide a probabilistic framework for generating subsequences with prescribed representation behavior, with potential impact on related additive problems.
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 $ψ$ is an increasing function tending to infinity with $ψ(x)=o(\log x)$ 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 $$ R_{s}(n;\mathfrak{X}_{k})\sim \mathfrak{S}(n)ψ(n)$$ 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.