An old  result of Kaufman showed that the set of badly approximable numbers supports a family of probabilty measures with polynomial decay rate on their Fourier transform. We show that the same phenomenon can be observed in a two-dimensional setup: Consider the set of pairs (alpha, gamma) in [0,1]^2 for which there exists c>0 such that all integers p,q satisfy

| q alpha - p - gamma | > c.

We prove that it supports certain probability measures with Frostman dimension arbitrarily close to $2$ and Fourier transform with polynomial decay rate. This is joint work with S. Chow and E. Zorin.