The $N$-measure for algebraic integers having all their conjugates in a sector

Abstract

Let $\alpha$ be a nonzero algebraic integer of degree $d$ whose all conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\dots ,{\alpha }_d$ lie in a sector $|argz|\le 𝜃$, $0\le 𝜃\le 90^{\circ }$. We define the N-measure of $\alpha$ by $N\left(\alpha \right)={\prod }_{i=1}^d\left(\left|{\alpha }_i\right|+1∕\left|{\alpha }_i\right|\right)$ and the absolute N-measure of $\alpha$ by $\nu \left(\alpha \right)=N{\left(\alpha \right)}^{1∕deg\left(\alpha \right)}$. Firstly, we consider the case $𝜃=0$. We prove that $N\left(\alpha \right)\in \mathbb{N}$ and that, if $\alpha$ is a reciprocal algebraic integer, $N\left(\alpha \right)$ is a square. Then, we study the set $\mathcal{𝒩}$ of the quantities $\nu \left(\alpha \right)$. We prove that there exists a number $l$ such that $\mathcal{𝒩}$ is dense in $\left(l,\infty \right)$. Finally, using the method of auxiliary functions, we find the seven smallest points of $\mathcal{𝒩}$ in $\left(2,l\right)$. In case of $0<𝜃\le 90^{\circ }$, we compute the greatest lower bound $c\left(𝜃\right)$ of the absolute N-measure of $\alpha$, for $\alpha$ belonging to eight subintervals of $$]