Fourier decay of self-similar measures

CAROLINA MOSQUERA; PABLO SHMERKIN

CABA

Congreso; CIMPA2017 Research School - IX Escuela SANTALÓ Harmonic Analysis, Geometric Measure Theory and Applications; 2017

The decay properties of the Fourier transform $\widehat{\nu}(\xi)$ of a finite measure $\nu$ as $|\xi|\to\infty$ give crucial``arithmetic'' information about $\mu.$ The Fourier transform $\widehat{\nu}(\xi)$ has {\it polynomial decay} if $|\widehat{\nu}(\xi)|\le C|\xi|^{-\sigma/2}$ for some constants $C_{\sigma}, \sigma>0.$ The supreme of such $\sigma$ is the {\it Fourier dimension } of $\nu.$The decay of Bernoulli convolutions (that is, self-similar measures for the iterated function systems $\{ax+t_i\}$ with weights $p_i$ and $a\in(0,1)$) has been studied in classic works of Erd\"{o}s \cite{Erd39, Erd40} and Kahane \cite{Kah71}.In this context, in this work we present results about the polynomial decay of the Fourier transform outside a small set of exceptions, giving explicit estimates. Also, we study different types of dimensions of Bernoulli convolutions.