An Integral Mean Value Theorem for Weyl Sums over Broken Arcs

Abstract

In this article, we research the mean value of integral of exponential sum $S(\alpha)=\sum_{u\in I}e(\alpha u^k)$, where $I$ is a short interval whose length is $2N^{\theta},\theta<1$, and the broken arc $\mathfrak{m^*}$ is a subset of a following minor arcs \[ \mathfrak{m}=\bigcap_{j\leq k-1}\left\{\alpha:\forall q<(\log N)^A,h<q,(h,q)=1,\Big|\alpha-\frac{h}{q}\Big|>\frac{1}{qN^{(k-j-1/2)\theta}}\right\} \] which has measure at least $c>0$. By setting $m$ is a sufficiently large number, $N$ is be sufficiently large in terms of $m$. When $k>\log m$ and $\frac{\log k}{\log m}<1/2$ we set the following estimate: \[ \int_{\mathfrak{m^*}}\bigg|\sum\limits_{{N_1}<u<{N_2}}{e(zu^k)}\bigg|^{m}\mathrm{d}z\ll_cN^{\theta m(\frac{2k+1}{2k+2}+o(1))} \] We can find this estimate moving beyond the even-odd restriction of powers. To get this bound, we first set a strong estimate for almost $\alpha$ by Diophantine approximation and Vinogradov's main value theorem. Then, combining this result, we construct a refined Weyl differencing argument by partitioning the differences step into large and small range, which significantly outperforms the classical one. By the version of probability, we can calculate the multiplicity of each sum. Put them together and we can complete the proof.