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

Abstract

Let α be a nonzero algebraic integer of degree d whose all conjugates α1=α,α2,…,αd lie in a sector |argz|≤𝜃, 0≤𝜃≤90∘. We define the N-measure of α by N(α)= ∏i=1d(|αi|+1∕|αi|) and the absolute N-measure of α by ν(α)=N(α)1∕ deg(α). Firstly, we consider the case 𝜃=0. We prove that N(α)∈ℕ and that, if α is a reciprocal algebraic integer, N(α) is a square. Then, we study the set 𝒩 of the quantities ν(α). We prove that there exists a number l such that 𝒩 is dense in (l,∞). Finally, using the method of auxiliary functions, we find the seven smallest points of 𝒩 in (2,l). In case of 0<𝜃≤90∘, we compute the greatest lower bound c(𝜃) of the absolute N-measure of α, for α belonging to eight subintervals of ]