# Poster Session/Reception

Tuesday, September 30, 2014 - 3:00pm - 4:30pm

Lind 400

**Lower Bound in the Roth Theorem for Amenable Groups**

Qing Chu (The Ohio State University)

We show a Khintchine-type extension of the Roth theorem for amenable groups. Namely, if $T_1, T_2$ are two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically, then $mu(Acap T_{1}^{g}A cap T_{1}^{g}T_{2}^{g}A) > mu(A)^{4} - epsilon$ on a syndetic set for any measurable set $A$ and any $epsilon>0$. The proof uses the concept of a sated system introduced by Austin.**An Ergodic Approach to Furstenberg-Katnelson-Weiss Type Results**

Tatchai Titichetrakun (University of British Columbia)

We are interested in the Furstenberg-Katnelson-type results in the discrete setting: Finding a congruent copy of a sufficiently large dilate of a finite configuration in a set of positive density in integer lattice of sufficiently large dimensions. The simplest case, when the configuration has just 2 points, is to say that such a set contains all sufficiently large distances. We will discuss an approach to distance problem using convolutions of surface measures, making kernels less singular and apply the idea of characteristic factors from ergodic theory. This avoids the use of spectral measure and seems more generalizable to more complicated configurations.**Asymptotically Approximate Groups**

Melvyn Nathanson (City University of New York)

If A_1,A_2,..., A_h are nonempty subsets of a group G, then their product set is

A_1 A_2 ... A_h = { x_1 x_2 ... x_h : x_i in A_i for i=1, 2,..., h }.

If A_i = A for all i=1,..., h, then we denote the product set A_1A_2 ... A_h by A^h.

A nonempty subset A of a group G is a K-approximate group if

there exists a subset X of G such that X leq K and A^2 subseteq XA.

We do not assume that A contains the identity, nor that A is symmetric,

nor that A is finite. The set A is an asymptotic K-approximate group

if the product set A^h is a K-approximate group for all sufficiently large h.

Theorem 1: Every finite set of integers is an asymptotic 3-approximate group.

Theorem 2: Every finite subset of the additive group Z^n of n-dimensional

lattice points is an asymptotic K-approximate group.

Theorem 3: Every finite subset of an abelian group is an asymptotic K-approximate group.

These results suggest the following questions.

Problem 1: Is every finite subset of the Heisenberg group H_3(Z) an asymptotic K-approximate group?

Problem 2: Is every finite subset of a nilpotent group an asymptotic K-approximate group?**A Rainbow Ramsey Analogue of Rado's Theorem**

Reuben La Haye (University of California)

We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. Specifically, we classify the rainbow regular matrices. A matrix with rational entries is rainbow regular if for all sufficiently large k, for all natural numbers n, for every equinumerous k-coloring of [kn], there exists a rainbow vector in the kernel of the matrix. The proof uses techniques from the Geometry of Numbers.