Tangent measures of typical measures

We prove that for a typical Radon measure mu in R^d, every non-zero Radon measure is a tangent measure of mu at mu almost every point. This was already shown by T. O'Neil in his PhD thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.Comment: v3: 20 pages, 4 figures, the main result was already proved by T. O'Neil in his PhD thesis with a different proof, added a reference to it. Peer-reviewed versio

Tangent measures of non-doubling measures

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.Comment: 17 pages, 5 figures. v2: Minor corrections throughout, and section six completely rewritten in a more reader-friendly style; Accepted to Math. Proc. Cambridge Philos. So

Trigonometric series and self-similar sets

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 some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. The rate of $\widehat{\mu}(\xi) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for random walks on $\mathbb{R}$.Comment: 18 pages, v2: improved the main theore

Fourier transform of self-affine measures

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $\Gamma$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $\mu = \sum p_j f_j \mu$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $\Gamma$ is connected real split Lie group in the Zariski topology, then $\widehat{\mu}(\xi)$ has a power decay at infinity. Hence $\mu$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $\Gamma$ and non-compactness of the image of $\Gamma$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehat{\mu}$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.Comment: v2: 27 pages, updated references. Accepted to Advances in Mat

Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. An

On Fourier analytic properties of graphs

We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of the fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension zero.Comment: 11 pages, 1 figure; references added and typos corrected in v2; to appear in Int. Math. Res. Not. IMR

Radial projections of rectifiable sets

We show that if no $m$-plane contains almost all of an $m$-rectifiable set $E \subset \R^{n}$, then there exists a single $(m - 1)$-plane $V$ such that the radial projection of $E$ has positive $m$-dimensional measure from every point outside $V$.Comment: 6 pages, 2 figures, typos corrected and added references. Accepted to Annales Academi{\ae} Scientiarum Fennic{\ae} Mathematic

Dimension, entropy, and the local distribution of measures

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local homogeneity, porosity and conical densities of measures.Comment: v2: 23 pages, 6 figures. Updated references. Accepted to J. London Math. So