Related papers: Jacques Peyri\`ere et les produits de Riesz
We prove analogs of Peyri\`ere's mutual singularity theorem for standard and generalized Riesz products on the unit sphere of $\mathbb{C}^n$, $n\ge 2$. As a corollary, we obtain an analog of Zygmund's dichotomy for the Riesz products under…
The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real…
Exposition of the relation between quasi-independent and Sidon sets, theorems of Pisier according to the method of Bourgain
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising…
We estimate the energy and Hausdorff dimensions of the Riesz products on the unit sphere of $\mathbb{C}^n$, $n\ge 2$. Also, we obtain similar results for the pluriharmonic measures on the torus.
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is…
We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq…
The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…
In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral…
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with…
We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form $RY + Z, $ where $R \subseteq (0,\infty)$ and $Y, Z \subseteq \mathbb{R}^d$. We prove a theorem on the Lebesgue measure and Hausdorff…
We discuss a natural extension of Gilles Pisier's approach to the study of measure concentration, isoperimetry and Poincar\'e-type inequalities. This approach allows one to explore counterparts of various results about Gaussian measure in…
The paper entitled "Well posedness of general cross-diffusion systems", by C. Choquet, C. Rosier, L. Rosier, J. Diff. Eq. 2021, is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any…
We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y=…
Can one characterise the Fourier decay of a product measure in terms of the Fourier decay of its marginals? We make inroads on this question by describing the Fourier spectrum of a product measure in terms of the Fourier spectrum of its…
In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally `special' Sidon in…
In recent study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, it is essential to study the regularity property of various stochastic objects. These stochastic objects are…
The Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles. Assuming the number of factors is large, one gets a measure on the space of…
Based on the properties of distributions and measures with discrete support, we investigate temperate almost periodic distributions on the Euclidean space and connection with their Fourier transforms. We also study relations between the…
We study the "Fourier symmetry" of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (1) A one-side extension of Frostman's theorem, which connects the rate of decay…