On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915491651518464 |
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| author | Kingsbury-Neuschotz, Nathaniel |
| author_facet | Kingsbury-Neuschotz, Nathaniel |
| contents | Hickman and Wright proved an $L^2$ restriction estimate for the parabola $Σ$ in $\mathbb{Z}/N\mathbb{Z}$ of the form
$$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6}$$
for all functions $f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C}$ and any $ε>0$, and that this bound is sharp when $N$ has a large square factor, and especially for $N = p^2$ for $p$ a prime. In contrast, Mockenhaupt and Tao proved in the special case $N = p$ the stronger estimate
$$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}.$$
We extend the Mockenhaupt-Tao bound to the case of squarefree $N$, proving
$$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4},$$
and discuss applications of this result to uncertainty principles and signal recovery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$ Kingsbury-Neuschotz, Nathaniel Classical Analysis and ODEs Combinatorics Number Theory Hickman and Wright proved an $L^2$ restriction estimate for the parabola $Σ$ in $\mathbb{Z}/N\mathbb{Z}$ of the form $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6}$$ for all functions $f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C}$ and any $ε>0$, and that this bound is sharp when $N$ has a large square factor, and especially for $N = p^2$ for $p$ a prime. In contrast, Mockenhaupt and Tao proved in the special case $N = p$ the stronger estimate $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}.$$ We extend the Mockenhaupt-Tao bound to the case of squarefree $N$, proving $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4},$$ and discuss applications of this result to uncertainty principles and signal recovery. |
| title | On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$ |
| topic | Classical Analysis and ODEs Combinatorics Number Theory |
| url | https://arxiv.org/abs/2509.09885 |