[Paper Review] Discrete restriction estimates of epsilon-removal type for kth-powers and k-paraboloids

Kevin Henriot, Kevin Hughes|arXiv (Cornell University)|Oct 13, 2016

Advanced Harmonic Analysis Research16 references3 citations

TL;DR

This paper establishes $0$-removal restriction estimates for discrete $k$-th power and $k$-paraboloid surfaces in $\mathbb{Z}^d$ using number-theoretic methods inspired by Bourgain's work on the squares and parabola. It achieves full supercritical range estimates for $k$-paraboloids when $d > k(k-2)$ and for $k$-th powers in the full supercritical range, removing the $N^\varepsilon$ loss in key cases via Weyl exponential sum and Vinogradov mean value bounds.

ABSTRACT

We obtain restriction estimates of $ε$-removal type for the set of $k$-th powers of integers, and for discrete $d$-dimensional surfaces of the form \[ \{ (n_1,\dots,n_d,n_1^k + \dotsb + n_d^k) \,:\, |n_1|,\dots,|n_d| \leq N \}, \] which we term '$k$-paraboloids'. For these surfaces, we obtain a satisfying range of exponents for large values of $d,k$. We also obtain estimates of $ε$-removal type in the full supercritical range for $k$-th powers and for $k$-paraboloids of dimension $d < k(k-2)$. We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain's works on the restriction theory of the squares and the discrete parabola.

Motivation & Objective

• To establish $0$-removal restriction estimates for discrete surfaces defined by $k$-th powers and $k$-paraboloids in $\mathbb{Z}^d$.
• To extend Bourgain's number-theoretic approach to higher-degree surfaces beyond the squares and parabola.
• To remove the $N^\varepsilon$ loss in restriction estimates for $k$-paraboloids in the supercritical range when $d > k(k-2)$.
• To obtain sharp estimates in the full supercritical range for $k$-th powers and for $k$-paraboloids with $d < k(k-2)$.
• To provide truncated restriction estimates of the form $\int_{|F_a| \geq N^{d/2 - \zeta}\|a\|_2} |F_a|^q \, dm \lesssim N^{dq/2 - K}\|a\|_2^q$ for $q > 2K/d$.

Proposed method

• Uses a discrete extension operator $F_a^{(\mathbf{P})}(\boldsymbol{\alpha}) = \sum_{\mathbf{n} \in [-N,N]^d} a(\mathbf{n}) e(\mathbf{P}(\mathbf{n}) \cdot \boldsymbol{\alpha})$ for polynomial surfaces $\mathbf{P}$.
• Applies Weyl's inequality and Vinogradov's mean value theorem to bound exponential sums $T(\alpha, \theta)$ over major and minor arcs.
• Employs the Hardy–Littlewood circle method with smooth weights and Poisson summation to analyze major arcs, using stationary phase and van der Corput estimates.
• Applies Hua's bound on Gauss sums: $|q^{-1} S(a,b;q)| \lesssim_{\varepsilon} q^{-1/k + \varepsilon}$ for $(a,q) = 1$.
• Uses the major arc bound $|T(\alpha,\theta)| \lesssim_{\varepsilon} q^{-1/k + \varepsilon} \min(N, |\beta|^{-1/k})$ under $|\beta| \lesssim q^{-1}N^{-(k-1)}$.
• Combines minor arc bounds from Vinogradov's mean value theorem with major arc estimates to derive $L^p$ restriction estimates without $N^\varepsilon$ loss.

Experimental results

Research questions

• RQ1 Can $0$-removal restriction estimates be established for the discrete $k$-th power surface $\{n^k : 1 \leq n \leq N\}$ in the full supercritical range?
• RQ2 For $k$-paraboloids $\{(n_1,\dots,n_d, n_1^k + \cdots + n_d^k)\}$, does the $\varepsilon$-removal hold when $d > k(k-2)$?
• RQ3 Can truncated restriction estimates of the form $\int_{|F_a| \geq N^{d/2 - \zeta}\|a\|_2} |F_a|^q \, dm \lesssim N^{dq/2 - K}\|a\|_2^q$ be proven for $q > 2K/d$?
• RQ4 Is the $N^\varepsilon$ loss removable in the critical restriction estimate for $k$-paraboloids when $d < k(k-2)$?
• RQ5 How do Weyl exponential sum estimates and Vinogradov mean value bounds contribute to removing the $\varepsilon$-loss in discrete restriction theory?

Key findings

• For $k$-th powers with $k \geq 3$, the paper achieves $\varepsilon$-removal estimates in the full supercritical range $p > 2k/d$ for $d \geq k$.
• For $k$-paraboloids with $d > k(k-2)$, the $\varepsilon$-removal estimate holds in the full supercritical range $q > 2K/d$, where $K = dk$.
• The minor arc bound $|T(\alpha,\theta)| \lesssim_{\varepsilon} N^{1 - 1/(k(k-1)) + \varepsilon}$ is established using the latest Vinogradov mean value theorem bounds.
• On major arcs, the bound $|T(\alpha,\theta)| \lesssim_{\varepsilon} q^{-1/k + \varepsilon} \min(N, |\beta|^{-1/k})$ is derived via Poisson summation and stationary phase.
• The $\varepsilon$-loss is removed for $k$-paraboloids in the supercritical range when $d > k(k-2)$, extending Bourgain's earlier results on the parabola.
• The truncated estimate $\int_{|F_a| \geq N^{d/2 - \zeta}\|a\|_2} |F_a|^q \, dm \lesssim N^{dq/2 - K}\|a\|_2^q$ holds for $q > 2K/d$ with $\zeta = \min(2^{1-k}, 1/(k(k-1)))$.