Related papers: Dimension Spectra of Lines
A non-Hermitean random matrix model proposed a few years ago has a remarkably intricate spectrum. Various attempts have been made to understand the spectrum, but even its dimension is not known. Using the Dyson-Schmidt equation, we show…
A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower…
The n-th symmetric product of a metric space is the set of its nonempty subsets with cardinality at most n, equipped with the Hausdorff metric. We prove that every symmetric product of the line is an absolute Lipschitz retract and admits a…
Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the…
Given a positive, decreasing sequence $a,$ whose sum is $L$, we consider all the closed subsets of $[0,L]$ such that the lengths of their complementary open intervals are in one to one correspondence with the sequence $a$. The aim of this…
We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two…
Given $s\in(1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\}$$ and $$\overline{B}_s[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=s\}.$$ The main goal of this paper is to study the $(\alpha,\beta)$-lineability/spaceability…
In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differential equation with constant coefficients, defined over an affinoid domain of the analytic affine line $A_k^{1,an}$. We show that it is a…
This paper extends the recently obtained complete and continuous map of the Lattice Isometry Space (LISP) to the practical case of dimension 3. A periodic 3-dimensional lattice is an infinite set of all integer linear combinations of basis…
This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex…
In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a "large" set of real functions the Lagrange and Markov Dynamical spectrum associated to these…
We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of…
We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$…
We prove that Hilbert space is distortable and, in fact, arbitrarily distortable. This means that for all lambda >1 there exists an equivalent norm |.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there exist x,y in Y…
We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even.
The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension…
The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in…
The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension…
The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…
We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric…