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Károly Simon

Publications and source records attributed to Károly Simon.

32 records · Page 2Linked to original sources
arXiv2018

Fractal Percolations

One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention probability is assigned with which we retain the cube. The interior of the discarded cubes contain no points from the Fractal percolation set. In the retained ones we repeat the process ad infinitum. The set that remains after infinitely many steps is the Fractal percolation set. The homogeneous case is when all of these probabilities are the same. Recently, there have been considerable developments in regards with the projection and slicing properties in the homogeneous case. In the first part of this note we give an account of some of these recent results and then we discuss the difficulties and provide some new partial results in the non-homogeneous case.

math.DS↗
arXiv2017

Interior of sums of planar sets and curves

Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\left(A+Γ\right)^{\circ}$, when $Γ$ is a piecewise $\mathcal{C}^2$ curve and $A\subset \mathbb{R}^2.$ To begin, we give an example of a very large (full-measure, dense, $G_δ$) set $A$ such that $\left(A+S^1\right)^{\circ}=\emptyset$, where $S^1$ denotes the unit circle. This suggests that merely the size of $A$ does not guarantee that $(A+S^1)^{\circ }\ne\emptyset$. If, however, we assume that $A$ is a kind of generalized product of two reasonably large sets, then $\left(A+Γ\right)^{\circ}\ne\emptyset$ whenever $Γ$ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of $C:=C_γ\times C_γ$, $γ\geq \frac{1}{3}$, pinned at any point of $C$ has non-empty interior, where $C_γ$ (see (1.1)) is the middle $1-2γ$ Cantor set (including the usual middle-third Cantor set, $C_{1/3}$). Our proof for the middle-third Cantor set requires a separate method. We also prove that $C+S^1$ has non-empty interior.

math.CA↗
arXiv2017

Large Deviation Multifractal Analysis of a Process Modeling TCP CUBIC

Multifractal characteristics of the Internet traffic have been discovered and discussed in several research papers so far. However, the origin of this phenomenon is still not fully understood. It has been proven that the congestion control mechanism of the Internet transport protocol, i.e., the mechanism of TCP Reno can generate multifractal traffic properties. Nonetheless, TCP Reno does not exist in today's network any longer, surprisingly traffic multifractality has still been observed. In this paper we give the theoretical proof that TCP CUBIC, which is the default TCP version in the Linux world, can generate multifractal traffic. We give the multifractal spectrum of TCP CUBIC traffic and compare it with the multifractal spectrum of TCP Reno traffic. Moreover, we present the multifractal spectrum for a more general model, where TCP CUBIC and TCP Reno are special cases. Our results also show that TCP CUBIC produces less bursty traffic than TCP Reno.

math.PR↗
arXiv2017

Singularity versus exact overlaps for self-similar measures

In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-$α$ projections of the natural measure of the Sierpiński carpet. We present more general one-parameter families of self-similar measures $ν_α$, such that the set of parameters $α$ for which $ν_α$ is singular is a dense $G_δ$ set but this "exceptional" set of parameters of singularity has zero Hausdorff dimension.

math.DS↗
arXiv2016

Projections of Fractal percolation constructed with inhomogeneous probabilities

In this paper we consider fractal percolation random Cantor sets $E$ on the plane constructed with non-homogeneous probabilities. We focus on the case when the probabilities are large enough to guarantee that the almost sure dimension of $E$ is greater than $1$. Under this assumption in the case of homogeneous (equal) probabilities it was proved by Rams and the first author that the orthogonal projection of $E$ contains an interval simultaneously in all directions. Moreover, Peres and Rams proved the stronger result that the orthogonal projection of the natural measure on $E$ to every line is absolutely continuous with Hölder-continuous density. We point out that in the case of non-homogeneous probabilities neither of the two previous assertions remain valid. However, we also prove that in the non-homogeneous case every line whose tangent is neither a rational nor a Liouville number is non-exceptional. That is, almost surely for all of these directions the projection of $E$ contains some interval and the projection of the natural measure has Hölder-continuous density.

math.DS↗
arXiv2015

On the dimension of self-affine sets and measures with overlaps

In this paper we consider diagonally affine, planar IFS $Φ=\left\{S_i(x,y)=(α_ix+t_{i,1},β_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.

math.DS↗
arXiv2013

Projections of fractal percolations

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals.

math.DS↗
arXiv2013

The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}

math.DS↗
arXiv2013

The dimension of projections of fractal percolations

\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If $\dim_{\rm H}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\geq 2$ one-dimensional fractal percolations.

math.DS↗
arXiv2013

Slicing the Sierpiński gasket

We investigate the dimension of intersections of the Sierpiński gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrand's theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension.

math.DS↗
arXiv2011

The algebraic difference of two random Cantor sets: The Larsson family

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

math.PR↗
arXiv2005

Absolute continuity for random iterated function systems with overlaps

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + λ_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $λ_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors $y_1,y_2,...$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $χ= E[\log(λY)]$ be the Lyapunov exponent. Assuming that $χ< 0$, we obtain a family of conditional measures $ν_y$ on the line, parametrized by $y = (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h > |χ|$, then $ν_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |χ|$, then the measure $ν_y$ is singular and has dimension $h/|χ|$ for a.e. $y$. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory.

math.DS↗