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.