We obtain new bounds for the integer Chebyshev constant of intervals [p/q, r/s] where p, q, r and s are non-negative integers such that qr - ps = 1. As a consequence of the methods used, we improve the known lower bound for the trace of totally positive algebraic integers.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

Let $M\left(\alpha \right)$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $ℂ$. Then its Lehmer constant$L\left(V\right)$ is inf $M{\left(\alpha \right)}^{1/deg\left(\alpha \right)}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L\left(V\right)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L\left(V\right)$, which we call the transfinite Lehmer constant $L_{\infty }\left(V\right)$.Also, we prove the converse to Langevin’s Theorem, which states that $L\left(V\right)\>1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_{\infty }\left(V\right)$.

Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates $\alpha ₁=\alpha ,...,{\alpha }_d$ are positive real numbers. We study the set ₂ of the quantities ${\left({\prod }_{i=1}^d{(1+\alpha {²}_i)}^{1/2}\right)}^{1/d}$. We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.

Suppose $A\in G{l}_d\left(\mathbb{Z}\right)$ has a 2-dimensional expanding subspace $E^u$, satisfies a regularity condition, called “good star”, and has $A^{*}\ge 0$, where $A^{*}$ is an oriented compound of $A$. A morphism $\theta$ of the free group on $\{1,2,\cdots ,d\}$ is called a non-abelianization of $A$ if it has structure matrix $A$. We show that there is a tiling substitution$\Theta$ whose “boundary substitution” $\theta =\partial \Theta$ is a non-abelianization of $A$. Such a tiling substitution $\Theta$ leads to a self-affine tiling of $E^u\sim {ℝ}^2$ with $A_u{:=A|}_{E_u}\in G{L}_2\left(ℝ\right)$ as its expansion. In the last section we find conditions on $A$ so...