Some Mizohata-Takeuchi-type estimate for exponential sums

TL;DR

This paper establishes a Mizohata-Takeuchi-type estimate for quadratic exponential sums, linking the integral of the sum's magnitude squared with geometric properties of weight functions and employing advanced harmonic analysis techniques.

Contribution

It introduces a novel estimate for quadratic exponential sums using the $TT^*$ method and circle method, extending Mizohata-Takeuchi-type bounds to this setting.

Findings

The integral of |G|^2 weighted by ω is controlled by the supremum of ω over tubes and the L^2 norm of coefficients.

The estimate is proved via analysis of superlevel set distributions of G.

The approach combines harmonic analysis techniques like the $TT^*$ method and circle method.

Abstract

Let $R^{\frac{1}{2}}$ be a large integer, and $\omega$ be a nonnegative weight in the $R$-ball $B_R=[0,R]^2$ such that $\omega(B_R)\le R$. For any complex sequence $\{a_n\}$, define the quadratic exponential sum \[ G(x,t)=\sum_{n=1}^{R^{\frac{1}{2}}} a_n e\big(\frac{n}{R^{\frac{1}{2}}} x+\frac{n^2}{R} t\big). \] It holds that \[ \int |G|^2 \omega \lessapprox \sup_{T}\omega(T)^{\frac{1}{2}}\cdot R \,\|a_n\|_{l^2}^2 \] where $T$ ranges over $R\times R^{\frac{1}{2}}$ tubes in $B_R$. The proof is established through exploring the distributions of superlevel sets of the $G$ function. It is based on the $TT^*$ method and the circle method.