On the unit distance problem

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known bound is $Cn^{\frac{4}{3}}$. We show that if the set under consideration is well-distributed and the fixed distance is much smaller than the diameter of the set, then the exponent $\frac{4}{3}$ is significantly improved. Corresponding results are also established in higher dimensions. The results are obtained by solving the corresponding continuous problem and using a continuous-to-discrete conversion mechanism. The degree of sharpness of results is tested using the known results on the distribution of lattice points dilates of convex domains. We also introduce the following variant of the Erd\H os unit distance problem: how many pairs of points from a set of size $n$ are separated by an integer distance? We obtain some results in this direction and formulate a conjecture

Maximal averages over flat radial hypersurfaces

We prove nearly sharp Orlicz space estimates for maximal averages over flat radial hypersurfaces in ${\Bbb R}^d$, $d \ge 3$

Falconer conjecture in the plane for random metrics

We prove the following variant of the Falconer conjecture in the plane. If the dimension of a compact planar set is greater than one, then the distance set with respect to almost every ellipse has positive Lebesgue measure

Fourier transform, L2L^2 restriction theorem, and scaling

We show, using a Knapp-type homogeneity argument, that the $(L^p, L^2)$ restriction theorem implies a growth condition on the hypersurface in question. We further use this result to show that the optimal $(L^p, L^2)$ restriction theorem implies the sharp isotropic decay rate for the Fourier transform of the Lebesgue measure carried by compact convex finite hypersurfaces

Distinct distances on a sphere

We study the Erdos distance conjecture on the unit sphere in three dimensions using Fourier analytic methods.Comment: Erdos distance conjecture on the sphere is investigated under a discrete energy assumptio

Pinned distance problem, slicing measures and local smoothing estimates

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a probability measure $\mu_F$ on $F$ such that for $\mu_F$-a.e. $y\in F$, (1) $\dim_{{\mathcal H}}(\Delta^y(E))\geq\beta$ if $\dim_{{\mathcal H}}(E) + \frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d - 1 + \beta$; (2) $\Delta^y(E)$ has positive Lebesgue measure if $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d$; (3) $\Delta^y(E)$ has non-empty interior if $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F) > d+1$. We also show that in the case when $\dim_{{\mathcal H}}(E)+\frac{d-1}{d+1}\dim_{{\mathcal H}}(F)>d$, for $\mu_F$-a.e. $y\in F$, $$\left\{t\in{\Bbb R} : \dim_{{\mathcal H}}(\{x\in E:|x-y|=t\}) \geq \dim_{{\mathcal H}}(E)+\frac{d+1}{d-1}\dim_{{\mathcal H}}(F)-d \right\}$$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$. In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples

Endpoint bounds for the non-isotropic Falconer distance problem associated with lattice-like sets

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$ is greater than $\frac{d}{2}$. He also produced an example, based on the integer lattice, showing that the exponent $\frac{d}{2}$ cannot be improved. In this paper we prove the Falconer distance conjecture for this class of sets based on the integer lattice. In dimensions four and higher we attain the endpoint by proving that the Lebesgue measure of the resulting distance set is still positive if the Hausdorff dimension of $S$ equals $\frac{d}{2}$. In three dimensions we are off by a logarithm. More generally, we consider $K$-distance sets $\Delta_K(S)=\{{|a-b|}_K: a,b \in S\}$, where ${|\cdot|}_K$ is the distance induced by a norm defined by a smooth symmetric convex body $K$ whose boundary has everywhere non-vanishing Gaussian curvature. We prove that our endpoint result still holds in this setting, providing a further illustration of the role of curvature in this class of problems.Comment: 13 page

Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $|\Delta(E)| \gtrsim q$ whenever $|E| \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. The second listed author and Misha Rudnev established the threshold $\frac{d+1}{2}$, and the authors of this paper, Doowon Koh and Misha Rudnev proved that this exponent is sharp in even dimensions. In this paper we improve the threshold to $\frac{d^2}{2d-1}$ under the additional assumption that $E$ has product structure. In particular, we obtain the exponent 4/3, consistent with the corresponding exponent in Euclidean space obtained by Wolff

Gabor orthogonal bases and convexity

Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ \{g(x-a)e^{2 \pi i x \cdot b} \}}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$

Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms

We prove that if $\phi: {\Bbb R}^d \times {\Bbb R}^d \to {\Bbb R}$, $d \ge 2$, is a homogeneous function, smooth away from the origin and having non-zero Monge-Ampere determinant away from the origin, then R^{-d} # \{(n,m) \in {\Bbb Z}^d \times {\Bbb Z}^d: |n|, |m| \leq CR; R \leq \phi(n,m) \leq R+\delta \} \lesssim \max \{R^{d-2+\frac{2}{d+1}}, R^{d-1} \delta \}. This is a variable coefficient version of a result proved by Lettington in \cite{L10}, extending a previous result by Andrews in \cite{A63}, showing that if $B \subset {\Bbb R}^d$, $d \ge 2$, is a symmetric convex body with a sufficiently smooth boundary and non-vanishing Gaussian curvature, then # \{k \in {\mathbb Z}^d: dist(k, R \partial B) \leq \delta \} \lesssim \max \{R^{d-2+\frac{2}{d+1}}, R^{d-1} \delta \}. (*) Furthermore, we shall see that the same argument yields a non-isotropic analog of $(*)$, one for which the exponent on the right hand side is, in general, sharp, even in the infinitely smooth case. This sheds some light on the nature of the exponents and their connection with the conjecture due to Wolfgang Schmidt on the distribution of lattice points on dilates of smooth convex surfaces in ${\Bbb R}^d$