On a Restriction Problem of Hickman and Wright for the Parabola over $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$

Abstract

Hickman and Wright proved an $L^2$ restriction estimate for the parabola $\Sigma$ over $\mathbb{Z}/N\mathbb{Z}$ of the form $${\left(\frac{1}{|\Sigma |}\sum _{m\in \Sigma }|\hat{f}(m){|}^2\right)}^{\frac{1}{2}}\le C_{\epsilon }{N}^{\epsilon }\cdot N^{-1}{\left(\sum _{x\in (\mathbb{Z}/N\mathbb{Z}{)}^2}|f(x){|}^{\frac{6}{5}}\right)}^{\frac{5}{6}}$$ for all functions $f:(\mathbb{Z}/N\mathbb{Z}{)}^2\to \mathbb{C}$ and any $\epsilon >0$, and showed that this bound is sharp when $N$ has a large square factor, especially for $N=p^2$ where $p$ is prime. In contrast, Mockenhaupt and Tao proved in the special case $N=p$ the stronger estimate $${\left(\frac{1}{|\Sigma |}\sum _{m\in \Sigma }|\hat{f}(m){|}^2\right)}^{\frac{1}{2}}\le C{N}^{-1}{\left(\sum _{x\in (\mathbb{Z}/N\mathbb{Z}{)}^2}|f(x){|}^{\frac{4}{3}}\right)}^{\frac{3}{4}}.$$ We extend the Mockenhaupt--Tao bound to the case of squarefree $N$, proving $${\left(\frac{1}{|\Sigma |}\sum _{m\in \Sigma }|\hat{f}(m){|}^2\right)}^{\frac{1}{2}}\le C_{\epsilon }{N}^{\epsilon }\cdot N^{-1}{\left(\sum _{x\in (\mathbb{Z}/N\mathbb{Z}{)}^2}|f(x){|}^{\frac{4}{3}}\right)}^{\frac{3}{4}},$$ and in fact a slightly sharper version with $C_{\epsilon }{N}^{\epsilon }$ replaced with $2^{\frac{\omega (N)}{4}}$, where $\omega (N)$ is the number of prime factors of $N$. We also discuss applications of this result to uncertainty principles and signal recovery.

10 pages. Corrected the proof of Lemma 2.2 to include the case of even N, unified notation, corrected typos, and tightened wording