Exact and Asymptotic Counts of MSTD, MDTS, and Balanced Sets in Dicyclic Groups

TL;DR

This paper provides exact and asymptotic counts of subsets with more sums than differences, more differences than sums, and balanced sets in Dicyclic groups, revealing their relationships and growth patterns.

Contribution

It determines exact counts for small subset sizes and asymptotic relationships for larger sizes in Dicyclic groups, including dependence on divisibility properties.

Findings

Exact counts for subsets of size two.

Asymptotic equality of MSTD and balanced subsets of size two.

Ratio of MSTD to MDTS subsets of size three for odd n.

Abstract

We investigate the relationship between the sizes of the sum and difference sets of the Dicyclic Group $\mathrm{Dic}_{4n}$. We first determine the exact numbers of MSTD (more sums than differences), MDTS (more differences than sums), and balanced subsets of size two. As a consequence, we show that the numbers of MSTD and balanced subsets of size two are asymptotically equal as $n \to \infty$. For odd $n$, we then obtain exact counts of MSTD, MDTS, and balanced subsets of size three, with the results depending on whether $n$ is divisible by $3$. In this case, we establish that asymptotically the number of MSTD subsets of size three is six times the number of MDTS subsets and also six times the number of balanced subsets. Finally, we establish a lower bound for the number of MSTD, MDTS, and balanced subsets of $\mathrm{Dic}_{4n}$ corresponding to the boundary case of size $2n$.