Abstract:  A result of Kaufman shows that the set ${\mathrm{\mathbf{Bad}}}$ of badly approximable numbers supports a family of probability measures with polynomial decay rate on their Fourier transform. We show that the same phenomenon can be observed in a two-dimensional setup: we consider the set 
\[ \mathcal{B} = \{ (\alpha, \gamma) \in [0,1]^2 : \inf \| q\alpha - \gamma\| > 0  \}  \]
and 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.