Restriction estimates of $$\varepsilon $$ ε -removal type for k-th powers and paraboloids
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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 $$\begin{aligned} \{ (n_1,\dots ,n_d,n_1^k + \cdots + n_d^k) \,:\, |n_1|,\dots ,|n_d| \leqslant N \}, \end{aligned}$$ 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.
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