Abstract
We obtain restriction estimates of varepsilon -removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form {(n1,⋯,nd,n1k+⋯+ndk):|n1|,⋯,|nd|⩽N},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\{ (n_1,\\dots ,n_d,n_1^k + \\cdots + n_d^k) \\,:\\, |n_1|,\\dots ,|n_d| \\leqslant N \\}, \\end{aligned}$$\\end{document}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 varepsilon -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.
Highlights
We are interested in restriction theorems for discrete surfaces in Zd
Where P = (P1, . . . , Pr ) is a system of r integer polynomials in d variables, and we assume that the map P : Zd → Zr is injective for simplicity
The surface we study is the truncated d-dimensional k-paraboloid embedded in Zd+1
Summary
We are interested in restriction theorems for discrete surfaces in Zd. We restrict our attention to parametric surfaces of the form. Corresponding to the system of polynomials P = (xk) of total degree k, when k 3 is an integer In this case we obtain the complete supercritical range of exponents for (1.5) and an incomplete range of exponents for truncated restriction estimates of the form (1.2). Improved bounds on Weyl sums are known for intermediate values of k, but they typically take a different shape than (1.8), and we do not try to incorporate them in our argument It seems that one current limitation of number-theoretic approaches to restriction estimates for surfaces of high degree is the poor quality of known minor arc bounds for Weyl sums. Results of ε-removal type ignore minor arcs to some extent, the efficient ranges in those cases
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