We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-1-7,
author = {Paulius Drungilas and Art\=uras Dubickas},
title = {Multiplicative dependence of shifted algebraic numbers},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {75-81},
zbl = {1041.11067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-1-7}
}
Paulius Drungilas; Artūras Dubickas. Multiplicative dependence of shifted algebraic numbers. Colloquium Mathematicae, Tome 96 (2003) pp. 75-81. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-1-7/