2022/10/03 by Ruben Ascoli, Ascoli, Ruben, Justin Cheigh +15 · 1 citation
Engineering · Mathematics · #05B10 #11P99 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2210.00669
openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a group G, we say that a set A ⊆ G has more sums than differences (MSTD) if |A+A| > |A-A|, has more differences than sums (MDTS) if |A+A| < |A-A|, or is sum-difference balanced if |A+A| = |A-A|. A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary finite groups G and proved that almost all subsets A⊆ G are sum-difference balanced as |G|→∞. For the dihedral group D2n, they conjectured that of the remaining sets, most are MSTD, i.e., there are more MSTD sets than MDTS sets. Some progress on this conjecture was made by Haviland et al. in 2020, when they introduced the idea of partitioning the subsets by size: if, for each m, there are more MSTD subsets of D2n of size m than MDTS subsets of size m, then the conjecture follows. We extend the conjecture to generalized dihedral groups D=ℤ2\ltimes G, where G is an abelian group of size n and the nonidentity element of ℤ2 acts by inversion. We make further progress on the conjecture by considering subsets with a fixed number of rotations and reflections. By bounding the expected number of overlapping sums, we show that the collection \mathcal SD,m of subsets of the generalized dihedral group D of size m has more MSTD sets than MDTS sets when 6≤ m≤ cj√(n) for cj=1.3229/√(111+5j), where j is the number of elements in G with order at most 2. We also analyze the expectation for |A+A| and |A-A| for A⊆ D2n, proving an explicit formula for |A-A| when n is prime.