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Sequences of algebraic integers and density modulo  1

Roman Urban — 2007

Journal de Théorie des Nombres de Bordeaux

We prove density modulo 1 of the sets of the form { μ m λ n ξ + r m : n , m ∈ ℕ } , where λ , μ ∈ ℝ is a pair of rationally independent algebraic integers of degree d ≥ 2 , satisfying some additional assumptions, ξ ≠ 0 , and r m is any sequence of real numbers.

Equidistribution in the dual group of the S -adic integers

Roman Urban — 2014

Czechoslovak Mathematical Journal

Let X be the quotient group of the S -adele ring of an algebraic number field by the discrete group of S -integers. Given a probability measure μ on X d and an endomorphism T of X d , we consider the relation between uniform distribution of the sequence T n 𝐱 for μ -almost all 𝐱 ∈ X d and the behavior of μ relative to the translations by some rational subgroups of X d . The main result of this note is an extension of the corresponding result for the d -dimensional torus 𝕋 d due to B. Host.

Unbounded harmonic functions on homogeneous manifolds of negative curvature

Richard Penney;  Roman Urban — 2002

Colloquium Mathematicae

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney;  Roman Urban — 2013

Studia Mathematica

Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to ℝ k , k>1. We consider a class of second order left-invariant differential operators on S of the form ℒ α = L a + Δ α , where α ∈ ℝ k , and for each a ∈ ℝ k , L a is left-invariant second order differential operator on N and Δ α = Δ - ⟨ α , ∇ ⟩ , where Δ is the usual Laplacian on ℝ k . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively) we obtain an...

Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h;  Roman Urban — 2005

Studia Mathematica

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ℝ d is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ℝ d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ℙ d - 1 . In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

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