Multiplicative dependence of the translations of algebraic numbers

Abstract

In this paper, we first prove that given pairwise distinct algebraic numbers $\alpha_1, \ldots, \alpha_n$, the numbers $\alpha_1+t, \ldots, \alpha_n+t$ 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 integers. For such a pair $(a,b)$ with $b-a=30$ we show that there are $13$ integers $t$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We conjecture that $13$ is the largest value of such translations for any $(a,b)$, where $a \ne b$, prove this for all pairs $(a,b)$ with difference at most $10^{10}$, and, assuming that the $ABC$ conjecture is true, show that for any such pair $(a,b)$, $a \ne b$, there is an absolute bound $C_1$ (independent of $a$ and $b$) on the number of such translations $t$.