Absolute continuity of non-homogeneous self-similar measures

Santiago Saglietti; Pablo Shmerkin; Boris Solomyak

doi:10.1016/j.aim.2018.06.015

Absolute continuity of non-homogeneous self-similar measures

Santiago Saglietti
, Pablo Shmerkin
, Boris Solomyak

Department of Mathematics

Technion-Israel Institute of Technology
Universidad Torcuato Di Tella

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

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.

Original language	English
Pages (from-to)	60-110
Number of pages	51
Journal	Advances in Mathematics
Volume	335
DOIs	https://doi.org/10.1016/j.aim.2018.06.015
State	Published - 7 Sep 2018

Bibliographical note

Publisher Copyright:
© 2018

Funding

B.S. was partially supported by the Israel Science Foundation grant 396/15.

Funders	Funder number
Consejo Nacional de Investigaciones Científicas y Técnicas	11220150100355
Israel Science Foundation	396/15, 1723/14

Keywords

Absolute continuity
Convolutions
Random measures
Self-similar measures

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10.1016/j.aim.2018.06.015

Cite this

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AU - Shmerkin, Pablo

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AB - 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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KW - Convolutions

KW - Random measures

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Absolute continuity of non-homogeneous self-similar measures

Abstract

Bibliographical note

Funding

Keywords

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