Related papers: Fourier transform of self-affine measures

We show that for Lebesgue almost all $d$-tuples $(\theta_1,\ldots,\theta_d)$, with $|\theta_j|>1$, any self-affine measure for a homogeneous non-degenerate iterated function system $\{Ax+a_j\}_{j=1}^m$ in ${\mathbb R}^d$, where $A^{-1}$ is…

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for…

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the…

Given any function $\phi \colon [0,\infty)\to (0,1]$ satisfying $\lim_{\xi\to\infty}\phi(\xi) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose…

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{\varphi_{\lambda}(x) = A_{\lambda}x + v_{\lambda}\}_{\lambda\in\Lambda}$ and a probability vector $p=(p_{\lambda})_{\lambda}>0$. Assume the strong…

We consider iterated function systems (IFS) in ${\mathbb R}^d$ for $d\ge 3$ of the form $\{f_j(x) = \lambda {\mathcal O} x + a_j\}_{j=0}^m$, with $a_0=0$ and $m\ge 1$. Here $\lambda\in (0,1)$ is the contraction ratio and ${\mathcal O}$ is…

Let $\mu$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $\Gamma_{\mu}$ generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique…

For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{\"o}s [2] and Salem [12]. We prove weak forms of their results for…

We establish a complete algebraic characterization of self-similar iterated function systems $\Phi$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure…

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost…

R. Kaufman and M. 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…

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support.…

We prove that the pushforwards of a very general class of fractal measures $\mu$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,\eta>0$ such…

Let $R$ be an expanding matrix with integer entries and let $B,L$ be finite integer digit sets so that $(R,B,L)$ form a Hadamard triple on ${\br}^d$ in the sense that the matrix $$ \frac{1}{\sqrt{|\det R|}}\left[e^{2\pi i \langle…

Let $ \mu $ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ \Phi = \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\in\Lambda} $ on $…

Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. 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,…

We study measures on $\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions.…

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\rho:\Gamma\to G_2$ and a surprising role played by higher rank…

In this paper, we investigate the Fourier transform of self-similar measures on R. We provide quantitative decay rates of Fourier transform of some self-similar measures. Our method is based on random walks on lattices and Diophantine…

Let $\nu$ be the Furstenberg measure associated with a non-elementary probability measure $\mu$ on SL_2(R). We show that, when $\mu$ has a finite second moment, the Fourier coefficients of $\nu$ tend to zero at infinity. In other words,…