Related papers: Extravagance, irrationality and Diophantine approx…

We study when Fourier transforms of Gibbs measures of sufficiently nonlinear expanding Markov maps decay at infinity at a polynomial rate. Assuming finite Lyapunov exponent, we reduce this to a nonlinearity assumption, which we verify for…

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for…

We classify mapping class group invariant probability measures on the character varieties of Deroin-Tholozan representations, namely the compact components of relative $\mathrm{PSL}_2\mathbb{R}$-character varieties. We prove that an ergodic…

We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schr\"odinger equation $i \partial_t u + \Delta u = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of…

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (R\'enyi). We show that they coincide exactly with two sets of quadratic…

In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $\psi:\mathbb{N}\to (0,1/2)$ with…

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…

The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in…

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

In a first part, using the recent measure classification results of Eskin--Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space $G/\Gamma$. Employing…

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but…

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus $\mathcal{T}_0$, invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian…

Let $h$ be a log-correlated Gaussian field on $\R^d$, let $\gamma \in (0,\sqrt{2d}),$ let $\mu_h$ be the $\gamma$-Gaussian multiplicative chaos measure, and let $D_h$ be an exponential metric associated with $h$ satisfying certain natural…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

We study the probabilistic behaviour of the continued fraction expansion of a quadratic irrational number, when weighted by some "additive" cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with…

As represented by the Liouville measure, Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. Under certain technical assumptions, we prove the convergence of a process time-changed by Gaussian multiplicative…

In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…