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We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and…
We investigate the geometry of approximates in multiplicative Diophantine approximation. Our main tool is a new multiparameter averaging result for Siegel transforms on the space of unimodular lattices in ${\mathbb R}^n$ which is of…
We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a…
We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\psi$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for…
We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local…
Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of…
We present a new unified theory of critical finite-size scaling for lattice statistical mechanical models with periodic boundary conditions above the upper critical dimension. Our theory is based on recent mathematically rigorous results…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…
In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff…
In this paper, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of…
In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journ\'ees EDP 2024. With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of…
We calculate the measure and Hausdorff dimension of sets of matrices over fields of formal power series with good approximation properties for a restricted set of denominators.
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
Spectral analysis plays a crucial role in high-dimensional statistics, where determining the asymptotic distribution of various spectral statistics remains a challenging task. Due to the difficulties of deriving the analytic form, recent…
We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings,…
When dealing with certain kind of complex phenomena the theoretician may face some difficulties -- typically a failure to have access to information for properly characterize the system -- for applying the full power of the standard…
This paper presents a brand new methodology to deal with isotopic fine structure calculations. By using the Poisson approximation in an entirely novel way, we introduce mathematical elegance into the discussion on the trade-off between…
We investigate superdiffusion for stochastic processes generated by nonuniformly hyperbolic system models, in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which…
This is a survey article describing some recent results at the interface of homogeneous dynamics and Diophantine approximation.
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of…