On Restriction Estimates for the Zero Radius Sphere over Finite Fields

Abstract

We study the finite field Fourier restriction/extension problem for spheres in even dimensions d ≥ 4. We prove that the L p → L 4 extension estimate for spheres of non-zero radii holds for 4d 3d−2 ≤ p ≤ ∞. Our result is sharp and improves the L (12d−8)/(9d−12)+ε → L 4 extension result for all ε > 0 due to the first and second listed authors [7]. The key new ingredient is improved additive energy estimates for subsets of spheres in even dimensions. In particular, our additive energy estimate improves and extends Rudnev's recent work [15] in four dimensions to higher even dimensions.As the most interesting result of this paper, we prove that if −1 is not a square number of F * q and the dimension d is 4k + 2 for some k ∈ N, then the L 2 → L (2d+4)/d extension estimate for spheres of zero radius holds. This result is also sharp and provides us of an interesting fact that the L 2 → L r extension estimate for zero spheres with specific assumptions is much better than the Stein-Tomas result which can not be improved in general for cones or zero spheres in even dimensions.2010 Mathematics Subject Classification. 52C10, 42B05, 11T23 .

An Asymmetric Bound for Sum of Distance Sets

“…We remark here that one can apply directly Theorem 1.3 in [5] for characteristic functions to give a bound which is better than (2.3), but weaker than (2.4).…”

A spherical extension theorem and applications in positive characteristic

“…If −1 is not a square and d = 4k +2, Iosevich, Lee, Shen, and the authors [11] proved that the Conjecture (1.4) holds for the sphere of zero radius. The main difference between the zero radius and non-zero radius spheres is that we can use the Gauss sum in the place of the Kloosterman sum in the Fourier decay.…”

The Spherical Kakeya Problem in Finite Fields

“…Spheres over finite fields are well-studied objects, see [7,8,12] and are defined as follows. Throughout this paper we assume that q is the power of an odd prime.…”