Multiplicative dependence of the translations of algebraic numbers

Abstract

In this paper, we first prove that given pairwise distinct algebraic numbers ${\alpha }_1,\dots ,{\alpha }_n$, the numbers ${\alpha }_1+t,\dots ,{\alpha }_n+t$ are multiplicatively independent for all sufficiently large integers $t$. Then, for a pair $(a,b)$ of distinct integers, we study how many pairs $(a+t,b+t)$ are multiplicatively dependent when $t$ runs through the set integers $\mathbb{Z}$. Assuming the $ABC$ conjecture we show that there exists a constant $C_1$ such that for any pair $(a,b)\in {\mathbb{Z}}^2$, $a\ne b$, there are at most $C_1$ values of $t\in \mathbb{Z}$ such that $(a+t,b+t)$ are multiplicatively dependent. For a pair $(a,b)\in {\mathbb{Z}}^2$ with difference $b-a=30$ we show that there are 13 values of $t\in \mathbb{Z}$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We further conjecture that 13 is the largest number of such translations for any such pair $(a,b)$ and prove this for all pairs $(a,b)$ with difference at most $10^{10}$.