Some Mizohata-Takeuchi-type estimate for exponential sums

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 $$G(x,t)=\sum _{n=1}^{R^{\frac{1}{2}}}{a}_{n}e(\frac{n}{R^{\frac{1}{2}}}x+\frac{n^2}{R}t)\mathrm{.}$$ It holds that $$\int \mathrm{\mid }G{\mathrm{\mid }}^2\omega ⪅\underset{T}{\sup }\omega (T{)}^{\frac{1}{2}}\cdot R\mathrm{\parallel }{a}_n{\mathrm{\parallel }}_{l^2}^2$$ 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 $T{T}^{\ast }$ method and the circle method.