Related papers: Approximation by random fractions
We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay…
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity,…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…
This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a…
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…
We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.
We study causal inference under case-control and case-population sampling. Specifically, we focus on the binary-outcome and binary-treatment case, where the parameters of interest are causal relative and attributable risks defined via the…
For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…
Inspired by a problem proposed by Mahler, we will address the following related question, 'How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To…
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…
Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…
Many AI researchers argue that probability theory is only capable of dealing with uncertainty in situations where a full specification of a joint probability distribution is available, and conclude that it is not suitable for application in…
Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence.…
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…
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…
This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…