Multiplicative dependence of two integers shifted by a root of unity

Abstract:

In this note we prove a result on the multiplicative independence of the numbers $m-\alpha ,n-\alpha$, where $m>n$ are positive integers and $\alpha$ is a reciprocal algebraic number with the property that $\alpha +1/\alpha$ has at least two real conjugates over $\mathbb {Q}$ lying in the interval $(-\infty ,2]$. As an application, we show that for any positive integers $m>n$ and $k \geqslant 3$ the numbers $m-\zeta _k, n-\zeta _k$, where $\zeta _k$ is the primitive $k$th root of unity, are multiplicatively independent except when $(n,k)=(1,6)$. This settles a recent conjecture of Madritsch and Ziegler.