Roots of unity and projective equivalence

TL;DR

This paper provides an elementary proof that two projectively equivalent finite sets of roots of unity have at most 18 elements, improving upon previous bounds and analyzing the limitations of existing methods.

Contribution

It introduces a new elementary proof for the maximal size of such sets and examines why prior methods do not achieve the tightest bounds.

Findings

Maximal size of such sets is at most 18.

Existing methods fail to reach the tight bound in maximal cases.

Provides insight into the limitations of Beukers and Smyth's approach.

Abstract

Existing results of Fu show that, if two finite sets of roots of unity are projectively equivalent by a projective automorphism that does not act bijectively on the set of all roots of unity, then these sets consist of at most 14 points. Moreover, Fu constructs the two possible maximal sets, which are unique up to projective equivalence. In this article, we give an elementary proof that the cardinality of two such sets is at most 18 using the methods of Beukers and Smyth. Moreover, we show precisely how their method fails to give the tightest bound in the maximal cases of Fu.