We introduce the notion of a quantitatively pleasant action on ZdZd, and show how the polynomial walks can be used to exhibit the quantitative pleasantness of certain natural actions on ZdZd. As a consequence, we derive that for every set AA of positive density in Z3Z3, there exists k<=k(d(A))k<=k(d(A)) such that {xy−z2|(x,y,z)∈A−A}{xy−z2|(x,y,z)∈A−A} contains kZkZ. We also present an elementary proof of the following  result: given sets E1,E2E1,E2 in ZZ of positive density there exists k<=k(d(E1),d(E2))k<=k(d(E1),d(E2)) such that (E1−E1)(E2−E2)(E1−E1)(E2−E2) contains kZkZ.

Based on a joint work with Kamil Bulinski.


Alexander Fish

Research Area

University of Sydney


Wed, 03/10/2018 - 2:00pm


RC-4082, The Red Centre, UNSW