Tomasz Downarowicz — Page 3

arXiv2008

Solvability of Rado systems in D-sets

Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of $D$-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.

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arXiv2006

The law of series

We prove a general ergodic-theoretic result concerning the return time statistic, which, properly understood, sheds some new light on the common sense phenomenon known as {\it the law of series}. Let \proc be an ergodic process on finitely many states, with positive entropy. We show that the distribution function of the normalized waiting time for the first visit to a small cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. This fact has the following interpretation: The occurrences of such a "rare event" $B$ can deviate from purely random in only one direction -- so that for any length of an "observation period" of time, the first occurrence of $B$ "attracts" its further repetitions in this period.

math.DS↗