Here we characterise, in a complete and explicit way, the relations of algebraic dependence over $\mathbb{Q}$ of complex values of Hecke-Mahler series taken at algebraic points ${\underline{u}}_1,...,{\underline{u}}_m$ of the multiplicative group ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$, under a technical hypothesis that a certain sub-module of ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$ generated by the ${\underline{u}}_i$’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.

We consider the behavior of the power series ${}_0\left(z\right)={\sum }_{n=1}^{\infty }{\mu }^2\left(n\right)z^n$ as z tends to $e\left(\beta \right)=e^{2\pi i\beta }$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then ${}_0\left(e\left(\beta \right)r\right)=O\left({(1-r)}^{-1/2-\epsilon }\right)$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that ${}_0\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-1+\delta }\right)$.

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real $y$ has algebraic degree $D\>1$, then the number $\#\left(\right|y|,N)$ of 1-bits in the expansion of $\left|y\right|$ through bit position $N$ satisfies$$\#\left(\right|y|,N)\>C{N}^{1/D}$$for a positive number $C$ (depending on $y$) and sufficiently large $N$. This in itself establishes the transcendency of a class of reals ${\sum }_{n\ge 0}1/2^{f\left(n\right)}$ where the integer-valued...

Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body $$\left|x_1\right|\left({\left|x_1\right|}^3+{\left|x_2\right|}^3+{\left|x_3\right|}^3\right)\le 1.$$ In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved...

P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.