On questions of Cassels and Drungilas-Dubickas

TL;DR

This paper affirms a question by Drungilas-Dubickas assuming standard conjectures on smooth numbers, providing evidence against the Dubickas Conjecture, and explores algebraic number dependence related to Cassels' problem.

Contribution

It offers new results under standard conjectures that support the Dubickas Conjecture's implications for the universality of the Hurwitz zeta-function and addresses Cassels' problem on algebraic number dependence.

Findings

Affirmative answer to Drungilas-Dubickas question under conjectures

Evidence against the Dubickas Conjecture

Results related to algebraic number dependence

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.