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arXiv · 1610.03984

Discrete restriction estimates of epsilon-removal type for kth-powers and k-paraboloids

Kevin Henriot·Kevin Hughes
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Abstract

We obtain restriction estimates of $\epsilon$-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 $\epsilon$-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.

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Kevin Henriot, Kevin Hughes. 2016-10-13. Discrete restriction estimates of epsilon-removal type for kth-powers and k-paraboloids. https://arxiv.org/abs/1610.03984

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