Searcharxiv⌕ Search

arXiv subjects

A. Iosevich

Publications and source records attributed to A. Iosevich.

47 records · Page 3Linked to original sources
arXiv2014

A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d) s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds new light on the classical approach to these problems. The key to our approach is a group action perspective which leads to natural and effective formulae related to the classical Mattila integral.

math.CA↗
arXiv2013

Group actions and geometric combinatorics in ${\mathbb F}_q^d$

In this paper we apply a group action approach to the study of Erd\H os-Falconer type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists $s_0(d)<d$ such that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, with $|E| \ge Cq^{s_0}$, then $|T^d_d(E)| \ge C'q^{d+1 \choose 2}$, where $T^d_k(E)$ denotes the set of congruence classes of $k$-dimensional simplices determined by $k+1$-tuples of points from $E$. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh (\cite{CEHIK12}) for $T^d_k(E)$ with $2 \leq k \leq d-1$. A non-trivial result for $T^2_2(E)$ in the plane was obtained by Bennett, Iosevich and Pakianathan (\cite{BIP12}). These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent $\frac{4}{3}$, previously established in \cite{CEHIK12} for the $q\equiv3\mbox{ mod }4$ case to the case $q\equiv1\mbox{ mod }4$, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in ${\mathbb F}_q^d$, where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.

math.CO↗
arXiv2011

Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting

In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of $A \cdot A+...+A \cdot A$, where $A$ is a subset of the real line of a given Hausdorff dimension, $A+A=\{a+a': a,a' \in A \}$ and $A \cdot A=\{a \cdot a': a,a' \in A\}$. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ${\Bbb R}^d$ is sufficiently large, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of the set of $k$-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates is also discussed.

math.CA↗
arXiv2006

Sum-product estimates in finite fields

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{1/2} \lesssim |A| \lesssim q^{7/10}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{3/2}}{q^{1/4}$.

math.CO↗
arXiv2004

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 $Δ(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $Δ(S)$ 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 $Δ_K(S)=\{{|a-b|}_K: a,b \in S\}$, 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.

math.CA↗
arXiv2003

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.

math.CA↗
arXiv2003

K-distance sets, Falconer conjecture, and discrete analogs

We prove a series of results on the size of distance sets corresponding to sets in the Euclidean space. These distances are generated by bounded convex sets and the results depend explicitly on the geometry of these sets. We also use a diophantine mechanism to convert continuous results into distance set estimates for discrete point sets.

math.CA↗
arXiv2003

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

We prove that the spherical mean of the Fourier transform of the characteristic function of a bounded convex set (without any additional assumptions) or a bounded set with a C^{3/2} boundary decays at infinity at the same rate as the Fourier transform of the characteristic function of the ball.

math.CA↗