Alladi Ramakrishnan Hall

Polynomial IP van der Waerden Theorem for Nilpotent Groups

Dibyendu De

University of Kalyani

A set of natural numbers is called syndetic if gaps in it are bounded
and called thik if it contains arbitrary long block. Further a set
will be piece wise syndetic if it can be expressed as an intersection
of

Using a dynamical approach, Furstenberg and Weiss extended van der
Waerden\textquoteright s theorem to arbitrary abelian groups and restricted
the arith- metic structure to IP-sets \cite{key-3}. In fact they
introduced the notion of IP mapping from the partial semigroup of
all finite subsets of set of natural numbers (takin union as semigroup
operation) to a commutative group. They proved that any piecewise
syndetic set contains contains IP progression of arbitrary length.

It is natural to ask if there are extensions of abovf Theorem to non-
abelian groups. However, in the case of nilpotent groups it is. By
interpreting IP map- pings as \textquotedblleft polynomial mappings
of degree 1\textquotedblright , Bergelson and Leibman in \cite{key-2=00005B2=00005D}
used this insight to prove a powerful polynomial extension of the
aboveTheorem for nilpotent groups.

Depending on their work we shall pose in this lecture some open problems
and conjecture.