Local entropy averages and projections of fractal measures (2009)
Hochman, Michael, Shmerkin, Pablo
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure...
On the dimension of iterated sumsets (2009)
Schmeling, Jörg, Shmerkin, Pablo
Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = A+...+A (k times). We show that for any non-decreasing sequence {a_k} taking...
Convolutions of Cantor measures without resonance (2009)
Nazarov, Fedor, Peres, Yuval, Shmerkin, Pablo
Denote by $\mu_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty \omega_j a^j$, where $P(\omega_j=0)=P(\omega_j=1)=1/2$ and all the choices are independent. For $0
The Hausdorff dimension of the projections of self-affine carpets (2009)
Ferguson, Andrew, Jordan, Thomas, Shmerkin, Pablo
We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a...
Local entropy averages and projections of fractal measures (2009)
Hochman, Michael, Shmerkin, Pablo
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure...
On the dimension of iterated sumsets (2009)
Schmeling, Jörg, Shmerkin, Pablo
Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = A+...+A (k times). We show that for any non-decreasing sequence {a_k} taking...
Convolutions of Cantor measures without resonance (2009)
Nazarov, Fedor, Peres, Yuval, Shmerkin, Pablo
Denote by $\mu_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty \omega_j a^j$, where $P(\omega_j=0)=P(\omega_j=1)=1/2$ and all the choices are independent. For $0
The Hausdorff dimension of the projections of self-affine carpets (2009)
Ferguson, Andrew, Jordan, Thomas, Shmerkin, Pablo
We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a...
Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets (2008)
We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor...
Overlapping self-affine sets of Kakeya type (2007)
Kaenmaki, Antti, Shmerkin, Pablo
We compute the Minkowski dimension for a family of self-affine sets on the plane. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this...
Resonance between Cantor sets (2007)
Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We...
The structure of overlapping self-affine sets / (2006)
Thesis (Ph. D.)--University of Washington, 2006.
A Modified Multifractal Formalism for a Class of Self-similar Measures with Overlap (2005)
The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H\left\{x:\lim_{r\rightarrow 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha\right\}. \] For...
Overlapping self-affine sets (2004)
We study families of possibly overlapping self-affine sets. Our main example is a family that can be considered the self-affine version of Bernoulli convolutions and was studied, in the...
A Modified Multifractal Formalism for a Class of Self-Similar Measures (2004)
The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures...