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arXiv · 1709.05092

Absolute continuity of non-homogeneous self-similar measures

Santiago Saglietti·Pablo Shmerkin·Boris Solomyak
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Abstract

We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.

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Santiago Saglietti, Pablo Shmerkin, Boris Solomyak. 2017-09-15. Absolute continuity of non-homogeneous self-similar measures. https://doi.org/10.1016/j.aim.2018.06.015

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