Quantitative growth of multi-recurrence sequences
Clemens Fuchs, Armand Noubissie
TL;DR
This paper provides explicit, quantitative bounds for the growth of multi-recurrence sequences over both number fields and function fields. By leveraging the quantitative Subspace theorem of Evertse in the number-field case and Brownawell–Masser plus Zannier’s function-field analogues in the function-field case, the authors derive concrete upper bounds on growth rates and valuations for multi-recurrences, extending FH’s and LV’s non-effective qualitative results to effective, computable estimates. The main contributions are Theorem 2 (number fields) and Theorem 9 (function fields), together with corollaries that sharpen and generalize prior results, and they address open questions posed in FH1. The results significantly advance the understanding of maximal growth behavior in multi-parameter linear recurrences, offering tools for explicit counting and growth control across arithmetic and geometric settings.
Abstract
In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we prove a quantitative version of that result by giving an explicit upper bound for the maximal possible growth rate of a multi-recurrence. Moreover, we also give a function field analogue of the result, answering a question posed by Fuchs and Heintze when proving a bound on the growth of multi-recurrences in number fields.