Some Mizohata-Takeuchi-type estimate for exponential sums
Xuerui Yang
TL;DR
This work proves a Mizohata-Takeuchi type bound for a quadratic exponential sum associated to the truncated parabola, under a one-dimensional weight in the plane. The key ingredients are a locally constant property for quadratic sums, sharp $L^4$ level-set control on a box decomposition, and a circle-method based local $L^2$ estimate that leverages an incidence geometry bound for rationals. The main result shows that $\int_{B_R} |G|^2 \omega \lesssim \sup_T \omega(T)^{1/2} \cdot R \|a\|_2^2$ for $G(x,t)=\sum_{n=1}^{R^{1/2}} a_n e(n/R^{1/2} x + n^2/R t)$ and $\omega(B_R) \le R$, with the proof combining TT$^*$ and circle-method techniques. The paper also exhibits level-set decay $\#_{\lambda} \lesssim N^2 \lambda^{-4}$ and provides counterexamples showing that the analogous bound fails for general convex $\Gamma$, highlighting the special role of the parabola and horizontal tubes in this setting.
Abstract
Let $R^{\frac{1}{2}}$ be a large integer, and $ω$ be a nonnegative weight in the $R$-ball $B_R=[0,R]^2$ such that $ω(B_R)\le R$. 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 \[ \int |G|^2 ω\lessapprox \sup_{T}ω(T)^{\frac{1}{2}}\cdot R \,\|a_n\|_{l^2}^2 \] 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.