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arXiv · 2607.15635

Smooth projections of self-similar measures

Amir Algom·Federico Rodriguez Hertz·Zhiren Wang
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Abstract

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.

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Amir Algom, Federico Rodriguez Hertz, Zhiren Wang. 2026-07-17. Smooth projections of self-similar measures. https://arxiv.org/abs/2607.15635

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