Formal-dual subsets of cyclic groups of prime power order

TL;DR

This paper investigates formal-duality in cyclic groups of prime power order, proving the non-existence of primitive dual subsets in most cases, thus supporting a conjecture about their uniqueness.

Contribution

It proves that cyclic groups of odd prime power order and certain even orders lack primitive formally-dual subsets, partially confirming a conjecture about their uniqueness.

Findings

No primitive formally-dual subsets in cyclic groups of odd prime power order.

No primitive formally-dual subsets in cyclic groups of order 2^{2l+1}.

Supports the conjecture that only trivial groups have such subsets.

Abstract

We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Sch\"urmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order $2^{2l+1}$, there is no primitive pair of formally-dual subsets. This partially proves a conjecture, made by the priorly mentioned authors, that the only cyclic groups with a pair of primitive formally-dual subsets are $\{0\}$ and $\mathbb{Z}/4\mathbb{Z}$.