On the Fourier decay and the dimension of self-similar measures

Abstract

In this thesis, entropy estimates and combinatorial methods are used to study the dimension and the speed of the decay of the Fourier transform of self-similar measures on $\R$. An iterated function system (IFS) on $\R$ is a collection of maps $f_1, \dots, f_r$ with $f_i(x)=\lambda_i x+\tau_i$, where $|\lambda_i|<1$ and $\lambda_i, \tau_i \in \R$. The self-similar measure $\mu$ associated to an IFS and to probability weights $p_1, \dots, p_r$ is the unique measure such that [ \mu=\sum_i p_i f_i^(\mu), ] where $f_i^(\mu)$ denotes the push forward of $\mu$ under $f_i$. %Studying properties, like dimension, absolute continuity and the Fourier decay of $\mu$, is one of the central objectives in the field of fractal geometry. It is expected that unless there are algebraic obstructions, self-similar measure have full dimension, are absolutely continuous with respect to the Lebesgue measure and have power Fourier decay, meaning that $|\widehat{\mu}(\xi)| \lsim |\xi|^{-\alpha}$ for some $\alpha>0$. While this was eventually proved for (Lebesgue) almost all parameters, little is known for any particular choice. In recent years, a quantiy called \textit{Garsia} (or \textit{random walk}) \textit{entropy} has emerged as a crucial concept to study self-similar measures. Roughly stated, the Garsia entropy quantifies how fast the number of distinct maps $f_{i_1} \circ \dots \circ f_{i_n}$ grows with $n$. The first self-similar measures to be studied were the so-called Bernoulli convolutions, associated to the IFS $\lambda x, \lambda x+1$ for $\lambda \in (0.5,1)$ with equal weights, considered by Erd\H os around 1940. %For those, %[ %\mu=\mathrm{law}\left(\sum_{j=0}^\infty \xi_j \lambda^j \right), %] %where $\xi_j$ are iid Bernoulli, that is, equidistributed on ${0, 1}$. In his 2019 break-through paper \cite{varjuannals}, Varj'u showed that Bernoulli convolutions have full dimension for all transcendental $\lambda$. A crucial ingredient of his proof was a result by Breuillard and Varj'u, relating the Garsia entropy of Bernoulli convolutions to the algebraic complexity of $\lambda$ \cite{bv}. The first chapter of this thesis considers the question when the Garsia entropy is maximal and classifies this occurrence. It is also shown that full dimension, absolute continuity and power Fourier decay (the strongest possible decay) are present in this case. The proof considers the question on sub-products of the adeles and uses the adelic Fourier transform. The second chapter contains mild progress towards generalising Varj'u's result from Bernoulli convolutions to all self-similar measures, which would solve the celebrated exact overlaps conjecture. The major roadblock preventing the resolution of this conjecture is the missing generalisation of \cite{bv} from Bernoulli convolutions to the general case. In Chapter \ref{chapter::curve}, the approach in \cite{bv} is strengthened to cover some further cases. The third chapter is concerned with the speed of Fourier decay for inhomogeneous IFS (i.e., not all $\lambda_i$ are the same). It is shown that if $\lambda_i$ are inverses of integers and $\tau_i$ are rational, then the speed of decay to expect is log-decay. More precisely, it is shown that unless an unexpected phenomenon occurs, which can be checked by finite computation, there is a sequence $\xi_n \to \infty$ such that $|\widehat{\mu}(\xi_n)| \sim \log^{-\beta}(|\xi_n|)$ for some $\beta>0$. %As an example, these computations are performed to show $\widehat{\mu}(6^n) \sim n^{-1/2}$ for the IFS $\frac{1}{2}x, \frac{1}{2}x+1, \frac{1}{3}x,\frac{1}{3}x+1$. Lastly, it is shown that a class of measures (for rational and some algebraic contractions) have power Fourier decay. These are the first examples in the inhomogeneous setting for which the precise decay is known. In the final chapter, certain random subsets of Cantor sets are studied. The problem was introduced in \cite{allaart-jones} where both an upper and lower bound for the dimension of these sets was shown. Here, the precise value of the dimension is calculated.