166 research outputs found
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 separated by a fixed (Euclidean)
distance is for any . The best
known bound is . 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 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 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 ,
Fourier transform, restriction theorem, and scaling
We show, using a Knapp-type homogeneity argument, that the
restriction theorem implies a growth condition on the hypersurface in question.
We further use this result to show that the optimal 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 we prove that for any , there
exists a probability measure on such that for -a.e. ,
(1) if ;
(2) has positive Lebesgue measure if ;
(3) has non-empty interior if .
We also show that in the case when , for -a.e. , has
positive Lebesgue measure. This describes dimensions of slicing subsets of ,
sliced by spheres centered at .
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
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 such that
whenever , where , the
-dimensional vector space over a finite field with elements (not
necessarily prime). Here . The second listed author and Misha Rudnev established the threshold
, 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 under the additional assumption that
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 be the indicator function of a bounded convex set in
, , with a smooth boundary and everywhere non-vanishing
Gaussian curvature. Using a combinatorial appraoch we prove that if , then there does not exist such that is an orthonormal basis for
Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms
We prove that if , , 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 , , 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
Group actions and a multi-parameter Falconer distance problem
In this paper we study the following multi-parameter variant of the
celebrated Falconer distance problem. Given with and , we define where for we write
with .
We ask how large does the Hausdorff dimension of need to be to ensure
that the -dimensional Lebesgue measure of is
positive? We prove that if for , then the
conclusion holds provided We
also note that, by previous constructions, the conclusion does not in general
hold if A group action derivation of a
suitable Mattila integral plays an important role in the argument
Erdos-Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields
We prove several incidence theorems in vector spaces over finite fields using
bounds for various classes of exponential sums and apply these to
Erdos-Falconer type distance problems
- β¦