Dirichlet uniformly well-approximated numbers

2015/08/03 by Kim, Dong Han, Liao, Lingmin · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1508.00520

Abstract

Fix an irrational number θ. For a real number τ>0, consider the numbers y satisfying that for all large number Q, there exists an integer 1≤ n≤ Q, such that ‖nθ-y‖0, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of θ. It is also proved that with respect to τ, the only possible discontinuous point of the Hausdorff dimension is τ=1.

Cited by

• Uniform Diophantine approximation on the plane for β-dynamical systems

Related

• Hausdorff dimension in inhomogeneous Diophantine approximation
• Inhomogeneous Diophantine approximation with general error functions
• A localized Jarnik-Besicovitch Theorem
• Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation
• Uniform Diophantine approximation related to beta-transformations