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 $$\left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_\epsilon N^\epsilon\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6}$$ for all functions $f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C}$ and any $\epsilon>0$, 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 $$\left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}.$$ We extend the Mockenhaupt--Tao bound to the case of squarefree $N$, proving $$\left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_\epsilon N^\epsilon\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4},$$ 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.