TuanTran

Let $X$ be a real random variable; a typical anti-concentration inequality asserts that (under certain assumptions) if an interval $I$ has small length, then $\mathbb{P}(X\in I)$ is small, regardless the location of $I$. Inequalities of this type have found powerful applications in many branches of mathematics. In this talk we will discuss several recent applications …

A family $\mathcal{F}$ of subsets of is called s-saturated if it contains no s pairwise disjoint sets, and moreover, no set can be added to $\mathcal{F}$ while preserving this property. More than 40 years ago, Erdős and Kleitman conjectured that an s-saturated family of subsets of has size at least $(1–2^{-(s-1)})2^n$. …

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of …

A theoretical dynamical system is a pair (X,T) where X is a compact metric space and T is a self homeomorphism of X. The topological entropy of a theoretical dynamical system (X,T), first introduced in 1965 by Adler, Konheim and McAndrew, is a nonnegative real number that measures the complexity of the system. Systems with positive …