Extension theorems for spheres in the finite field setting

Abstract

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields. In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, previous results by the authors in [6]. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.

On some problems of Euclidean Ramsey theory

“…P r o o f. We will prove just (6), the proof of (4), (5) is similar and is contained in [6], Lemma 2. Put S = S j .…”

“…Also we consider a model situation of the plane over the prime finite field F p × F p and prove (a slightly stronger) analog of Theorem 1, see section 2. The proof develops the method from [1], [6], [12].…”

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

“…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.…”

“…The first result in Proposition 1.1 was given in [6] and it gives the Stein-Tomas result, the L 2 → L (2d+2)/(d−1) bound. It is well known in [7] that the Stein-Tomas result gives the optimal L 2 → L r estimate for spheres in general odd dimensions. In even dimensions d, it is conjectured that the "r" index of the Stein-Tomas result can be improved to (2d + 4)/d.…”

Extension theorems for paraboloids in the finite field setting

“…See, for example, [6,7,9]. Mockenhaupt and Tao [9] first posed the extension problems in the finite field setting for various algebraic varieties S and they obtained reasonably good results from extension problems for paraboloids in d-dimensional vector spaces over finite fields.…”