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S-units and D-finite power series

Banff International Research Station for Mathematical Innovation and Discovery

Let $K$ be a field of characteristic zero and let $G$ be a finitely generated subgroup of $K^*$. Given a $P$-recursive sequence $f(n)$ taking values in $K$, we study the problem of when $f(n)$ takes values in $G$. We show that this problem can be interpreted purely dynamically and, when one does so, one can prove a much more general result about algebraic dynamical systems. Using this framework, we can then show that the set of $n$ for which $f(n)\in G$ is a finite union of infinite arithmetic progressions along with a set of zero Banach density, which simultaneously generalizes a result of Methfessel and a separate result due to Bezivin.

Mathematics; Combinatorics; Sequences, series, summability; Discrete mathematics

video/mp4

Author affiliation: University of Waterloo

2018-03-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65033

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/