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

TL;DR

This work investigates sumset and difference set sizes in the nonabelian dicyclic groups $\mathrm{Dic}_{4n}$, focusing on more-sums-than-differences (MSTD), more-differences-than-sums (MDTS), and balanced subsets. The authors develop a detailed combinatorial framework to count MSTD, MDTS, and balanced subsets by cardinality $k$, yielding exact results for $k=2$ and, for odd $n$, exact counts for $k=3$ that depend on whether $3\mid n$, along with asymptotic relations $\mathrm{S}_{\mathrm{Dic}_{4n}}(3) \sim 6\mathrm{D}_{\mathrm{Dic}_{4n}}(3)$ and $\mathrm{S}_{\mathrm{Dic}_{4n}}(3) \sim 6\mathrm{B}_{\mathrm{Dic}_{4n}}(3)$. The paper also provides nontrivial lower bounds in the boundary case $|A|=2n$, where the exponents of the rotation part form an arithmetic progression, via constructive families and residue-class analyses. These results reveal a distinct behavior in the dicyclic groups compared to dihedral groups and raise open questions about divisibility patterns and extensions to larger subset sizes. The methods combine explicit group-structure identities with careful case analyses of collision conditions in $A+A$ and $A-A$, yielding precise enumerative formulas and asymptotics.

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 $2n$.

Theorems & Definitions (36)

• Definition 1.1
• Conjecture 1.2
• Lemma 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Corollary 1.1
• Theorem 1.8
• Theorem 1.9