Some Mizohata-Takeuchi-type estimate for exponential sums
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| Format: | Preprint |
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2025
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| _version_ | 1866918181986107392 |
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| author | Yang, Xuerui |
| author_facet | Yang, Xuerui |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Mizohata-Takeuchi-type estimate for exponential sums Yang, Xuerui Classical Analysis and ODEs Number Theory 42B37 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. |
| title | Some Mizohata-Takeuchi-type estimate for exponential sums |
| topic | Classical Analysis and ODEs Number Theory 42B37 |
| url | https://arxiv.org/abs/2511.00841 |