ABSOLUTE CONTINUITY FOR RANDOM ITERATED FUNCTION SYSTEMS WITH OVERLAPS (2009)
Yuval Peres, Károly Simon, Boris Solomyak
Abstract. We consider linear iterated function systems with a random mul-tiplicative error on the real line. Our system is {x ↦ → di + λiY x} m i=1, where di ∈ R and λi> 0 are fixed and...
Fractals with Positive Length and Zero Buffon Needle Probability (2009)
Yuval Peres, Károly Simon, Boris Solomyak
polygonal approximation, is not useful for measuring the “length ” of more complicated sets. For a Borel (e.g., closed or open) subset F of R, the Lebesgue measure |F | of F,
On the Size of the Algebraic Difference of Two Random Cantor Sets* (2006)
ABSTRACT: In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies...
Visibility for self-similar sets of dimension one in the plane (2005)
Simon, Károly, Solomyak, Boris
We prove that a purely unrectifiable self-similar set of finite 1-dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point.
Absolute continuity for random iterated function systems with overlaps (2005)
Peres, Yuval, Simon, Károly, Solomyak, Boris
We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $\lambda_i>0$ are...
The absolute continuity of the distribution of randum sums with digits {0, 1, … , m-1} (2004)
Simon, Károly, Tóth, Hajnal R.
Let $m\geq 2$ be a natural number. Let $\nu _\lambda^{m} $ be the distribution of the random sum $\sum\limits_{n=0}^{\infty } \theta_n\lambda ^n$, where $\theta _n$ are i.i.d. and for every $n$ the...
The Equivalence Of Some Bernoulli Convolutions To Lebesgue Measure (1998)
R. Daniel Mauldin, Károly Simon
. Since the 1930's many authors have studied the distribution of the random series Y = P \Sigma n where the signs are are chosen independently with probability (1=2; 1=2) and 0 ! ! 1. Solomyak...