Multiplicative dependence of the translations of algebraic numbers

TL;DR

This paper proves that for large enough shifts, algebraic numbers become multiplicatively independent, and investigates the maximum number of translations where pairs of integers remain multiplicatively dependent, providing bounds and conjectures.

Contribution

It establishes the multiplicative independence of shifted algebraic numbers for large shifts and analyzes the maximum multiplicative dependence among translated integer pairs, including bounds and conjectural limits.

Findings

Algebraic numbers become multiplicatively independent after large shifts.

For pairs of integers with a fixed difference, the number of multiplicatively dependent translations is bounded.

Under the ABC conjecture, a uniform bound exists on such translations for all pairs.

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 $t$.