Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({\mathbb{R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({\mathbb{R}})$ matrices induces a (generalized) iterated function system on the projective line ${\mathbb{RP}}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

Dirichlet Uniformly Well-approximated Numbers

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∥ < Q−τ, where ∥⋅∥ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any τ > 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.

The Matiyasevich Theorem. Preliminaries

Summary In this article, we prove selected properties of Pell’s equation that are essential to finally prove the Diophantine property of two equations. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.

Lattice orbit distribution on ℝ2

We study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.

Majorant series convergence for twistless KAM tori

A self-contained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ‘strong diophantine property’ hypothesis used previously.