Restriction estimates of $$\varepsilon $$ ε -removal type for k-th powers and paraboloids

Abstract

We obtain restriction estimates of -removal type for the set of k -th powers of integers, and for discrete d -dimensional surfaces of the form 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 -removal type in the full supercritical range for k -th powers and for k -paraboloids of dimension . 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.

Discrete restriction estimates for forms in many variables

“…See [19,20] for a variant. Bourgain's approach has been abstracted in [22] and [17]. We combine Lemmas 3.3 and 3.6 from [17] to form the following lemma.…”

“…It transpired that Magyar’s question was partially answered in [17] where minor arc estimates were incorporated to prove discrete restriction estimates for ‘ k -paraboloids’. While [17] were predominately interested in ϵ -removal lemmas, the methods therein also used minor arc estimates to prove discrete restriction estimates.…”

“…It transpired that Magyar’s question was partially answered in [17] where minor arc estimates were incorporated to prove discrete restriction estimates for ‘ k -paraboloids’. While [17] were predominately interested in ϵ -removal lemmas, the methods therein also used minor arc estimates to prove discrete restriction estimates. When one observes that the worst error term (in Magyar’s generalizations of Magyar–Stein–Wainger’s decomposition) arises from the minor arcs, the natural strategy becomes to adapt those methods to handle the other error terms.…”

“…We organized our argument to closely follow [17] so that Theorem 1 reduces to proving appropriate estimates for the main term and the error term, but we streamline the approach to fit our purposes. In particular, since we are interested in the sharp discrete restriction estimates, our approach ‘bakes in’ the ϵ -removal.…”