Related papers: Logarithm laws and shrinking target properties
We study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules. We compare them with respect to the emergence of power laws in the size distribution of their components. We show…
We study the problem of approximating a discrete probability distribution, such as the next-token distribution of a large language model, by a dyadic distribution induced by a binary tree under encoding rate constraints. The objective is to…
We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…
This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…
We discuss the allocation of finite resources in the presence of a logarithmic diminishing return law, in analogy to some results from Information Theory. To exemplify the problem we assume that the proposed logarithmic law is applied to…
Projection algorithms are well known for their simplicity and flexibility in solving feasibility problems. They are particularly important in practice due to minimal requirements for software implementation and maintenance. In this work, we…
We establish higher order approximations in the Dynkin--Lamperti theorem, a limit theorem for the distribution of a killed subordinator immediately before its first passage time over a fixed level. For this purpose, we also study asymptotic…
In this paper we propose a wide class of truncated stochastic approximation procedures with moving random bounds. While we believe that the proposed class of procedures will find its way to a wider range of applications, the main motivation…
Fine-grained complexity theory is the area of theoretical computer science that proves conditional lower bounds based on the Strong Exponential Time Hypothesis and similar conjectures. This area has been thriving in the last decade, leading…
Mathematical diffraction theory is concerned with the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra…
In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior…
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the…
Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…
In order to bring contraction analysis into the very fruitful and topical fields of stochastic and Bayesian systems, we extend here the theory describes in \cite{Lohmiller98} to random differential equations. We propose new definitions of…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…
We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and…
Let \Gamma be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the points at infinity of the tree that are well approximated by the parabolic fixed points of G. Using Bruhat-Tits…