Some Mizohata-Takeuchi-type estimate for exponential sums

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.