Polynomial configurations in sets of positive density over local fields

A famous theorem of Furstenberg and Sarközy states that if f(x) ∈ Z[x] and f(0) = 0, then for every set I ⊂eq Z of positive upper density contains distinct elements x and y with x-y=f(n). The question of finding polynomial configurations of similar type can also be posed for general polynomials and sets of positive density over local fields. In this talk we will discuss some results in this direction and as some of their applications to estimating the Borel chromatic number of certain algebraically defined Cayley graphs of Rn or Qpn. This talk is based on two joint works with M. Bardestani.