Undecidability of expansions of Laurent series fields by cyclic discrete subgroups

Abstract: In 1987, Pheidas showed that the field of Laurent series $\mathbb{F}_q((t))$ with a constant for the indeterminate $t$ and a predicate for the natural powers ${t^n \mid n > 0}$ of $t$ is existentially undecidable. We show that the same result holds true if $t$ is replaced by any element $\alpha$ of positive $t$-adic valuation.