Ekstr\"om-Persson conjecture regarding random covering sets

We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekstr\"om and Persson concerning the exact value of the dimension in the special case of radii $(n^{-\alpha})_{n=1}^\infty$. For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekstr\"om-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.Comment: 25 pages, 1 figur

Entry and Return times distribution

This is a review article on the distributions of entry and return times in dynamical systems which discusses recent results for systems of positive entropy.Comment: To appear in "Dynamical Systems: An International Journal dedicated to the Statistical Properties of Dynamical Systems

Discrepancy convergence for the drunkard's walk on the sphere

We analyze the drunkard's walk on the unit sphere with step size theta and show that the walk converges in order constant/sin^2(theta) steps in the discrepancy metric. This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at http://www.math.hmc.edu/~su/papers.htm

Harmonic functions on locally compact groups of polynomial growth

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.Comment: 20 page