arXiv subjects

A. Greenleaf

Publications and source records attributed to A. Greenleaf.

4 recordsLinked to original sources

arXiv2024

Similar point configurations via group actions

We prove that for $d\ge 2,\, k\ge 2$, if the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$ is greater than $\frac{d^2}{2d-1}$, then, for any given $r > 0$, there exist $(x^1, \dots, x^{k+1})\in E^{k+1}$, $(y^1, \dots, y^{k+1})\in E^{k+1}$, a rotation $\theta \in \mathrm{O}_d(\mathbb{R})$, and a vector $a \in \mathbb{R}^d$ such that $rx^j = \theta y^j - a$ for $1 \leq j \leq k+1$. Such a result on existence of similar $k$-simplices in thin sets had previously been established under a more stringent dimensional threshold in Greenleaf, Iosevich and Mkrtchyan \cite{GIM21}. The argument we are use to prove the main result here was previously employed in Bhowmik and Rakhmonov \cite{BR23} to establish a finite field version. We also show the existence of multi-similarities of arbitrary multiplicity in $\R^d$, show how to extend these results from similarities to arbitrary proper continuous maps, as well as explore a general group-theoretic formulation of this problem in vector spaces over finite fields.

math.CA↗

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↗

arXiv2008

Approximate quantum cloaking and almost trapped states

We describe families of potentials which act as approximate cloaks for matter waves, i.e., for solutions of the time-independent Schrödinger equation at energy $E$, with applications to the design of ion traps. These are derived from perfect cloaks for the conductivity and Helmholtz equations, by a procedure we refer to as isotropic transformation optics. If $W$ is a potential which is surrounded by a sequence $\{V_n^E\}_{n=1}^\infty$ of approximate cloaks, then for generic $E$, asymptotically in $n$ (i) $W$ is both undetectable and unaltered by matter waves originating externally to the cloak; and (ii) the combined potential $W+V_n^E$ does not perturb waves outside the cloak. On the other hand, for $E$ near a discrete set of energies, cloaking {\it per se} fails and the approximate cloaks support wave functions concentrated, or {\it almost trapped}, inside the cloaked region and negligible outside. Applications include ion traps, almost invisible to matter waves or customizable to support almost trapped states of arbitrary multiplicity. Possible uses include simulation of abstract quantum systems, magnetically tunable quantum beam switches, and illusions of singular magnetic fields.

quant-ph↗

arXiv2008

Isotropic transformation optics: approximate acoustic and quantum cloaking

Transformation optics constructions have allowed the design of electromagnetic, acoustic and quantum parameters that steer waves around a region without penetrating it, so that the region is hidden from external observations. The material parameters are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. We address this problem by showing how to construct {\sl isotropic and nonsingular} parameters that give {\sl approximate} cloaking to any desired degree of accuracy for electrostatic, acoustic and quantum waves. The techniques used here may be applicable to a wider range of transformation optics designs. For the Helmholtz equation, cloaking is possible outside a discrete set of frequencies or energies, namely the Neumann eigenvalues of the cloaked region. For the frequencies or energies corresponding to the Neumann eigenvalues of the cloaked region, the ideal cloak supports trapped states; near these energies, an approximate cloak supports {\sl almost trapped states}. This is in fact a useful feature, and we conclude by giving several quantum mechanical applications.

physics.optics↗