Extension theorems for paraboloids in the finite field setting

In this paper we study the L ^p ? L ^r boundedness of the extension operators associated with paraboloids in ${{\mathbb F}_{q}^{d}}In this paper we study the L ^p − L ^r boundedness of the extension operators associated with paraboloids in \mathbb F_q^d{{\mathbb F}_{q}^{d}} , where \mathbbF_q{\mathbb{F}_{q}} is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x,y,z,w) ? E⁴{(x,y,z,w) \in E^4} with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L ^p to L ⁴ in the case when −1 is a square number in \mathbbF_q{\mathbb{F}_{q}} . Using the sharp L ^p −L ⁴ result, we improve upon the range of exponents r, for which the L ² − L ^r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in \mathbbF_q{\mathbb{F}_{q}}, we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.     

Distance Measures for Well-Distributed Sets

In this paper we investigate the Erdos/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions, we link this problem to the question about the existence of smooth well-curved hypersurfaces that support many integer points.     

Distance sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ? ?⁺ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [?q, q]²) ≦ C(K) _q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L ²(K) does not possess an orthogonal basis of exponentials.     

Random fluctuations of convex domains and lattice points

In this paper, we examine a random version of the lattice point problem. Let denote the class of all homogeneous functions in of degree one, positive away from the origin. Let be a random element of , defined on probability space , and define

for . We prove that, if , where , then

where , the expected volume. That is, on average, . We give explicit examples in which the Gaussian curvature of is small with high probability, and the estimate holds nevertheless.

     

Sharp rate of average decay of the Fourier transform of a bounded set

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L ²-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in $\mathbb{R}^{d}$ is $${\bigg(\int_{S^{d-1}}{\big|\widehat{\chi}_B(R\omega)\big|}^2d\omega \bigg)}^{{1}/{2}} \lesssim R^{-\frac{d+1}{2}}.\eqno(*)$$ This estimate is optimal for any convex body and in particular it agrees with the familiar estimate for the ball. The above estimate was proved in two dimensions by Podkorytov, and in all dimensions by Varchenko under additional smoothness assumptions. The main result of this paper proves (*) in all dimensions under the convexity hypothesis alone. We also prove that the same result holds if the boundary of is C^3/2.     

Spherical means and the restriction phenomenon

Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2) set σ_Θ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ², the inequality.

Average decay estimates for Fourier transforms of measures supported on curves

We consider Fourier transforms of densities supported on curves in ℝ^d. We obtain sharp lower and close to sharp upper bounds for the decay rates of as R → ∞.     

On sets of directions determined by subsets of ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Given E ? ? ^d , d ≥ 2, define

$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d ? 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d?1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. This result is sharp, since the conclusion fails to hold if E is a (d ? 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ? ^d . We also discuss the case when the Hausdorff dimension of E is precisely d ? 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d ? 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set P ? ? ^d , d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ? #P distinct directions.

     

Mean lattice point discrepancy bounds,II: Convex domains in the plane

We consider planar curved strictly convex domains with very weak (or no) smoothness assumptions on the boundary and prove sharp bounds for square-functions associated to the lattice point discrepancy. Research supported in part by NSF grants.