Arithmetic Progressions in the Primitive Length Spectrum

TL;DR

This paper proves that certain arithmetic locally symmetric orbifolds have primitive length spectra containing arbitrarily long arithmetic progressions, confirming a conjecture linking this property to the spaces' arithmetic nature.

Contribution

It establishes that all primitive lengths in these orbifolds appear in arbitrarily long arithmetic progressions, confirming a key conjecture about their characterization.

Findings

Primitive length spectrum contains arbitrarily long arithmetic progressions.

Every primitive length occurs in arbitrarily long arithmetic progressions.

Confirms the conjecture relating arithmeticity to length spectrum properties.

Abstract

In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progressions in its primitive length spectrum. This confirms one direction of a conjecture of Lafont--McReynolds, which states that the property of having every primitive length occur in arbitrarily long arithmetic progressions characterizes the arithmeticity of such spaces.