On questions of Cassels and Drungilas-Dubickas
arXiv:1606.02524
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 Kačinskaitė and Laurinčikas 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.
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