A Euclidean Fourier-analytic approach to vertical projections in the Heisenberg group

An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot-Carath\'eodory metric. This improves the known lower bound for sets $A$ with $1< \dim A < 7/2$, and answers a question of F\"assler and Hovila. The approach uses the Euclidean Fourier transform.Comment: 10 page