Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 823-834

SELF-SIMILAR MEASURES: ASYMPTOTIC BOUNDS FOR THE DIMENSION AND FOURIER DECAY OF SMOOTH IMAGES

Carolina A. Mosquera and Pablo S. Shmerkin

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales
Departamento de Matemática, Ciudad Universitaria, Pabellón I (C1428EGA)
Ciudad de Buenos Aires, Argentina and IMAS-CONICET, Argentina; mosquera 'at' dm.uba.ar

Torcuato di Tella University and CONICET, Department of Mathematics and Statistics
Av. Figueroa Alcorta 7350 (C1428BCW), Ciudad de Buenos Aires, Argentina; pshmerkin 'at' utdt.edu

Abstract. Kaufman and Tsujii proved that the Fourier transform of self-similar measures has a power decay outside of a sparse set of frequencies. We present a version of this result for homogeneous self-similar measures, with quantitative estimates, and derive several applications: (1) non-linear smooth images of homogeneous self-similar measures have a power Fourier decay, (2) convolving with a homogeneous self-similar measure increases correlation dimension by a quantitative amount, (3) the dimension and Frostman exponent of (biased) Bernoulli convolutions tend to 1 as the contraction ratio tends to 1, at an explicit quantitative rate.

2010 Mathematics Subject Classification: Primary 28A78, 28A80.

Key words: Fourier decay, self-similar measures, correlation dimension.

Reference to this article: C. A. Mosquera and P. S. Shmerkin: Self-similar measures: asymptotic bounds for the dimension and Fourier decay of smooth images. Ann. Acad. Sci. Fenn. Math. 43 (2018), 823-834.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4350