Norm form equations with solutions taking values in a multi-recurrence

TL;DR

This paper investigates solutions to norm form equations constrained by multi-recurrences, establishing finiteness results except for specific exceptional recurrence shapes, thus contributing to the understanding of solutions in structured sequences.

Contribution

It provides a finiteness theorem for solutions of norm form equations within multi-recurrences, identifying exceptional cases where infinite solutions may occur.

Findings

Finitely many solutions in each component unless recurrence has a specific shape

Characterization of exceptional recurrence shapes allowing infinite solutions

Extension of arithmetic progression results to norm form equations

Abstract

We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the recurrence is of precisely described exceptional shape. This gives a variant of the question on arithmetic progressions in the solution set of norm form equations.