In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, P...

The theory of Hausdorff dimension provides a general notion of the size of a set in a metric space. We define Hausdorff measure and dimension, enumerate some techniques for computing Hausdorff dimension, and provide applications to self-similar sets and Brownian motion. Our approach follows that of Stein [4] and Peres [3].

We settle an open problem, raised by Y. Peres and D. Revelle, concerning the L mixing time of the random walk on the lamplighter graph. We also provide general bounds relating the entropy decay of a Markov chain to the separation distance of the chain, and show that the lamplighter graphs once again provide examples of tightness of our results.

It is well-known that eigenvalues in quantum mechanics cannot be assigned to physical properties independently of the measurement context. We argue it might possible relate contexts Mermin-Peres magic square using causality relations a way which makes explanations hidden variable models unnecessary and unappealing.

Nazarov, Peres and Volberg proved recently in [8] that the Favard length of the n-th iteration of the four-corner Cantor set is bounded from above by n−c for an appropriate c. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

We use the evolving set methodology of Morris and Peres to show Cheeger inequalities for bounding the spectral gap of a finite ergodic Markov kernel. This leads to sharp versions of several previous inequalities, including ones involving edge-expansion and/or vertex-expansion. A bound on the smallest eigenvalue also follows

We show that every entanglement with positive partial transpose may be constructed from an indecomposable positive linear map between matrix algebra.

A 1965 problem due to Ludwig Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume 1. recent approach this is concerned the construction dense forests and obtained by suitable weakening constraint. forest discrete point getting uniformly close long enough line segments. The distribution points then quantified terms visibility functi...