Pure Point Dynamical and Diffraction Spectra (2009)
Lee, Jeong-Yup, Moody, Robert V., Solomyak, Boris
We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems (2009)
Lee, Jeong-Yup, Moody, Robert V., Solomyak, Boris
There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite...
On the topology of sums in powers of an algebraic number (2009)
Sidorov, Nikita, Solomyak, Boris
Let $1
SPECTRA OF BERNOULLI CONVOLUTIONS AS MULTIPLIERS IN L p ON THE CIRCLE (2009)
Nikita Sidorov, Boris Solomyak
Abstract. It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution µθ parameterized by a Pisot number θ is countable. Combined with results of Salem and Sarnak,...
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...
L q Densities for Measures Associated with Parabolic IFS with Overlaps (2009)
Boris Solomyak, Mariusz Urbański
ABSTRACT. We study parabolic iterated function systems (IFS) with overlaps on the real line and measures associated with them. A Borel probability measure µ on the coding space projects into a...
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,
PSEUDO-SELF-AFFINE TILINGS IN R d (2009)
Abstract. It is proved that every pseudo-self-affine tiling in R d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived...
Spacings and pair correlations for finite Bernoulli convolutions (2008)
Benjamini, Itai, Solomyak, Boris
We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the...
Quasisymmetric conjugacy between quadratic dynamics and iterated function systems (2008)
Eroğlu, Kemal Ilgar, Rohde, Steffen, Solomyak, Boris
We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ``overlap set'' $\Ok$ is finite, and which are ``invertible'' on the attractor $A$, the...
ERRATA TO “SPECTRUM OF DYNAMICAL SYSTEMS ARISING FROM DELONE SETS” (2008)
Boris Solomyak, Boris Solomyak
not satisfy the triangle inequality. There we used for two Delone sets Λ1 and Λ2, denoting by Br the ball of radius r centered at the origin: ρ(Λ1, Λ2) = min{2 −1/2, �ρ(Λ1, Λ2)}, where...
On the characterization of expansion maps for self-affine tilings (2008)
Kenyon, Richard, Solomyak, Boris
We consider self-affine tilings in $\R^n$ with expansion matrix $\phi$ and address the question which matrices $\phi$ can arise this way. In one dimension, $\lambda$ is an expansion factor of a...
Abstract. The distribution of the random series P (2007)
Yuval Peres, Wilhelm Schlag, Boris Solomyak
is the infinite convolution product
HOW LIKELY IS BUFFON'S NEEDLE TO FALL NEAR A PLANAR CANTOR SET? (2007)
Dedicated to the memory of Thomas H. Wol Abstract. Let be a compact planar set of positive nite one-dimensional Hausdor measure. Suppose that the intersection of with any rectiable curve has zero...
Boris Solomyak, Mariusz Urbanski, M. Urba Nski
. We continue to study parabolic iterated function systems (IFS) with overlaps on the real line and invariant measures associated with them. A shift-invariant Borel probability measure on the coding...
On the Context-Freeness of the Theta-Expansions of the Integers (2007)
Christiane Frougny, Boris Solomyak
Let ` ? 1 be a non-integral real number such that the `-expansion of every positive integer is finite. If the set of `-expansions of all the positive integers is a context-free language, then ` must...
The natural extension of the fi-transformation (2007)
Karma Dajani, Cor Kraaikamp, Boris Solomyak
For each real number fi? 1 the fi-transformation is defined by T fi x = fix(mod1). In this paper the natural extension T fi of the ergodic system underlying T fi is explicitly given. Furthermore, it...
SPECTRA OF BERNOULLI CONVOLUTIONS AS MULTIPLIERS IN L p ON THE CIRCLE (2007)
Nikita Sidorov, Boris Solomyak
Abstract. It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution µθ parameterized by a Pisot number θ, is countable. Combined with results of Salem and...
Equivalence of positive Hausdorff measure and the open set condition for self--conformal sets (2007)
Yuval Peres, Micha L Rams, K Aroly Simon, Boris Solomyak, Let V R
Abstract. A compact set K is self-conformal if it is a nite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the \open set condition...
Benjamini, Itai, Gurel-Gurevich, Ori, Solomyak, Boris
We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0 (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure of its interior; (3) for Pisot $1/b$ the...
Eigenfunctions for substitution tiling systems (2005)
We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous...
Pure point diffractive substitution Delone sets have the Meyer property (2005)
Lee, Jeong-Yup, Solomyak, Boris
We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J. C. Lagarias. We also show that for primitive substitution Delone...
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.
Pseudo-self-affine tilings in R^d (2005)
It is proved that every pseudo-self-affine tiling in R^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoi...
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...
Topological mixing for substitutions on two letters (2004)
Kenyon, Richard, Sadun, Lorenzo, Solomyak, Boris
We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution...
The sharp Hausdorff measure condition for length of projections (2004)
In a recent paper, Pertti Mattila asked which gauge functions $\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $\phi$, the projection of $A$ to...
Spectra of Bernoulli convolutions as multipliers in Lp on the circle (2003)
Sidorov, Nikita, Solomyak, Boris
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of R. Salem and P. Sarnak,...
Spectra of Bernoulli convolutions as multipliers in Lp on the circle (2003)
Sidorov, Nikita, Solomyak, Boris
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of R. Salem and P. Sarnak,...
Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle (2002)
Sidorov, Nikita, Solomyak, Boris
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $\mu_\theta$ parameterized by a Pisot number $\theta$, is countable. Combined with results of Salem and...
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions (2002)
We consider the ``Mandelbrot set'' $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters...
Characterization Of Planar Pseudo-Self-Similar Tilings (2001)
Natalie Priebe, Boris Solomyak
. A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a non-exact substitution rule: the substitution for a tile is not geometrically similar to itself. An example is...
Approximation By Polynomials With Coefficients (2000)
Yuval Peres And, Yuval Peres, Boris Solomyak
. In response to a question of R. Kenyon, we prove that the set of polynomials with coecients 1, evaluated at a xed real number , is dense in R for a.e. 2 ( p 2; 2). For 2 (1; p 2], a more complete...
Problems on self-similar sets, self-affine sets: an update (1999)
Self-similar sets that satisfy a certain separation condition (the open set condition) are quite well understood. However, self-similar sets with "overlap", and self-affine sets (with or...
Hausdorff dimension for horseshoes in R³ (1999)
Using pressure formulas we compute the Hausdorff dimension of the basic set of `almost every' C
Parabolic Iterated Function Systems With Overlaps I: Dimension Of The Limit Set. (1998)
K. Simon, Boris Solomyak, M. Urbanski, M. Urba Nski
. We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every...
Parabolic Iterated Function Systems With Overlaps II: Invariant Measures (1998)
K. Simon, Boris Solomyak, Mariusz Urbanski, M. Urba Nski
. We continue to study parabolic iterated function systems with overlaps on the real line. In [?] we investigated the Hausdorff dimension of the limit set; here we focus on the properties of...
Measure and Dimension for some Fractal Families (1998)
We study self-similar sets with overlaps, on the line and in the plane. It is shown that there exist self-similar sets that have non-integral Hausdorff dimension equal to the similarity dimension,...
Correlation dimension for self-similar Cantor sets with overlaps (1997)
In this paper we consider self-similar Cantor sets ae R which are either homogeneous and \Gamma is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of...
Self-Similar Measures And Intersections Of Cantor Sets (1997)
. It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g....
Non-Periodicity Implies Unique Composition for Self-Similar Translationally-Finite Tilings (1997)
Let T be a translationally-finite self-similar tiling of R d . We prove that if T is nonperiodic, then it has the unique composition property. More generally, T has the unique composition property...
Spectrum of dynamical systems arising from Delone sets (1997)
We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of...
Absolute Continuity of Bernoulli Convolutions (1996)
. The distribution of the random series P \Sigma n has been studied by many authors since the two seminal papers by Erdos in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba'nski,...
Dynamics of Self-Similar Tilings (1996)
This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of R d : The main focus is on spectral properties of...
Hausdorff Dimension Of Limit Sets For Parabolic Ifs With Overlaps
K. Simon, Boris Solomyak, Mariusz Urbanski, M. Urba Nski
. We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every...