This paper is concerned with the Diophantine properties of the orbits of any given point
under beta-transformations as beta varies. More precisely, let T β be the beta-transformation …

The most versatile version of the classical divergence Borel-Cantelli lemma shows that for
any divergent sequence of events $ E_n $ in a probability space satisfying a quasi …

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error
term. Precisely, let $\psi:\mathbb {N}\to [0, 1/2] $ be a function such that the series $\sum …

Let. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and
Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic …

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality
and intermittency of the family of generalized Riemann's non-differentiable functions R x 0 …

Abstract Let (Ω, A, μ) be a probability space. The classical Borel–Cantelli Lemma states that
for any sequence of μ-measurable sets E i (i= 1, 2, 3,…), if the sum of their measures …

In this paper we initiate a new approach to studying approximations by rational points to
points on smooth submanifolds of $\mathbb {R}^ n $. Our main result is a convergence …

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine
approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's …

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 …

Dirichlet's theorem, including the uniform setting and asymptotic setting, is one of the most
fundamental results in Diophantine approximation. The improvement of the asymptotic …