Improved Weyl bounds on short intervals

Abstract

For an integer $d\ge 3$, put ${\Delta }_d=\min 2^{d-1},d(d-1)$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}{X}^d+{\alpha }_{d-1}{X}^{d-1}+\cdots +{\alpha }_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|{\sum }_{n\in \mathcal{I}}e(P(n))\right|{\ll }_{d,\epsilon }{q}^{1/d}{H}^{\epsilon }+H^{1-1/{\Delta }_d+\epsilon }$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in \mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum n\in \mathcal{I}{e}_p(P(n))\right|{\ll }_{d,\epsilon }{H}^{1-\min u/d,1/{\Delta }_d+\epsilon }$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.

19 pages, no figures. Comments are welcome. Formalization project: https://github.com/hxypqr/improved-weyl-bounds-short-intervals