Description
A variant of Kaufman’s measures in two dimensions.
An old result of Kaufman showed that the set 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
B={(α,γ)∈[0,1]2 :inf∥qα−γ∥>0} and we prove that it supports certain probability measures with Frostman dimension arbitrarily close to 2 and Fourier transform with polynomial decay rate. (Joint work with S. Chow and E. Zorin).
