## Seminar

**Mathematics**

**Speaker**:

**Michael Boshernitzan**
Rice University (Mathematics)

### Geometry-Analysis Seminar: Some Combinatorics and Dynamical Systems connections

**
Wednesday, April 5, 2017 **

4:00 PM
to 5:00 PM
427 Herman Brown Hall

Rice University

6100 Main St

Houston, Texas, USA

Some combinatorial problems (some are open) with probability flavor will be discussed. Possible subjects: 1. We show that the collection $2^\N$ of subsets of natural numbers forms a complete space relative to the pseudometric \dist^*(U,V)=d^*(U\oplus V), U,V\subset \N, where $U\oplus V$ denotes the symmetric difference of $U, V$ and d^*(W):=\limsup_{n\to\infty}\,\frac{\big|W\cap[1,n]\big|}n, n\in\N, stands for the upper asymptotic density of $W\subset\N$. The above claim extends to some other notions ofdensity but not to the upper Banach density d_B^*(W):=\limsup_{n\to\infty},\frac{\big|W\cap[m,m+n]\big|}n, m,n\in\N. 2. We prove that every compact connected subset in $\R^2$ which does not reduce to a point must contain either a convex curve or an arithmetical progression of length 3.