TY - JOUR

T1 - Pointwise normality and Fourier decay for self-conformal measures

AU - Algom, Amir

AU - Rodriguez Hertz, Federico

AU - Wang, Zhiren

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/12/24

Y1 - 2021/12/24

N2 - Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

AB - Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

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U2 - 10.1016/j.aim.2021.108096

DO - 10.1016/j.aim.2021.108096

M3 - Article

AN - SCOPUS:85119054638

SN - 0001-8708

VL - 393

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108096

ER -