Smooth projections of self-similar measures

We prove a Furstenberg-type criterion for a given orthogonal projection of a self-similar measure to be absolutely continuous, with quantified regularity.

It requires exponential mixing of the rotational part at a rate that is sufficiently fast compared with an orbit relative analogue of its dimension.

Using Ramanujan sets of irrational rotations in \(\mathrm{SO}(3)\) constructed by Lubotzky, Phillips and Sarnak (1986, 1987), we obtain explicit applications.

In particular, we exhibit singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension with smooth projections in all but a fully explicit exceptional set of directions, and a non-trivial example of a self-similar measure that is Salem with a $C^2 _0$ density.