On questions of Cassels and Drungilas-Dubickas
Abstract
We answer a question of Drungilas-Dubickas in the affirmative under the assumption of standard conjectures on smooth numbers in polynomial sequences. This gives evidence against the "Dubickas Conjecture", which Ka\v{c}inskait\.e and Laurin\v{c}ikas proved implies universality results for the Hurwitz zeta-function with certain algebraic irrational parameters. Under these standard conjectures we also prove some results that confirms observations of Worley relating to a problem of Cassels on the multiplicative dependence of algebraic numbers shifted by integers.
Keywords
Cite
@article{arxiv.1606.02524,
title = {On questions of Cassels and Drungilas-Dubickas},
author = {Johan Andersson},
journal= {arXiv preprint arXiv:1606.02524},
year = {2016}
}
Comments
10 pages, A version of this manuscript was circulated in a smaller group in July 2011. When looking at the manuscript this year I decided that it might have some interest for a wider circle. The current manuscript is very similar to the manuscript from 5 years ago, although I have rewritten the proofs to hold for algebraic numbers rather than algebraic integers