Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands

Abstract

We obtain a uniform pointwise estimate for short Weyl sums of degree $n\geq3$ in the intermediate rational-approximation range $$ \frac{1}{qx^{n-2}y}\ll|\lambda| \ll\frac{1}{qy^{n-1}}. $$ This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For $r=2^n+1$ and fixed positive numbers $\mu_1,\ldots,\mu_r$ satisfying $$ \mu_1+\cdots+\mu_r=1, $$ we derive an asymptotic formula for the number of representations $$ x_1^n+\cdots+x_r^n=N, \qquad |x_i^n-\mu_iN|\leq H,\qquad 1\le i \le r, $$ valid for $$ N^{1-\theta(n,r)+\varepsilon}\le H \le \frac{N}{\ln N}, \quad \theta(n,r)= \frac{2}{n\bigl((r-1)(n-1)+2\bigr)}. $$ The resulting admissible lower bound for $H$ improves the previously known bound for every $n\geq3$.