论文标题
总体二面体组中的总和和差异集
Sum and Difference Sets in Generalized Dihedral Groups
论文作者
论文摘要
给定一个组$ g $，我们说如果$ | a+a | > | a-a | $，如果$ | a+a | <| a-a | $，或者如果$ | a+a |是sum-Difference = | a-a | $。最近的兴趣问题是了解这些类型的子集的频率。 第七作者和Vissuet研究了任意有限组$ g $的问题，并证明几乎所有子集$ a \ subseteq g $ sum-Difference均为$ | g | \ to \ infty $。对于二面体组$ d_ {2n} $，他们猜想其余集合中的大多数是MSTD，即，MSTD集比MDTS集更多。 Haviland等人在这种猜想上取得了一些进展。在2020年，当他们介绍按大小对子集进行分区的想法时：如果，对于每个$ m $，MSTD的$ d_ {2n} $的$ $ m $的MSTD子集比$ M $的MDTS子集的$ d_ {2n} $，则猜想在下面。 我们将猜想扩展到广义的二面体组$ d = \ mathbb {z} _2 \ ltimes g $，其中$ g $是一个尺寸$ n $的亚伯群，是$ \ mathbb {z} _2 _2 $的非身份元素。我们通过考虑具有固定数量旋转和反射数量的子集在猜想上进一步进步。通过限制预期的重叠总和的预期数量，我们表明，$ \ Mathcal s_ {d，m} $的广义二二二二二组$ d $ j $ m $的MSTD套件比$ 6 \ le m \ le m \ le c_j \ sqrt { $ c_j = 1.3229/\ sqrt {111+5j} $，其中$ j $是$ g $中的元素数量，最多为$ 2 $。我们还分析了$ | a+a | $和$ | a-a | $ for $ a \ subseteq d_ {2n} $的期望，证明了$ | a-a | $的明确公式。
Given a group $G$, we say that a set $A \subseteq G$ 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 $A\subseteq G$ are sum-difference balanced as $|G|\to\infty$. For the dihedral group $D_{2n}$, 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 $D_{2n}$ of size $m$ than MDTS subsets of size $m$, then the conjecture follows. We extend the conjecture to generalized dihedral groups $D=\mathbb{Z}_2\ltimes G$, where $G$ is an abelian group of size $n$ and the nonidentity element of $\mathbb{Z}_2$ 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 $\mathcal S_{D,m}$ of subsets of the generalized dihedral group $D$ of size $m$ has more MSTD sets than MDTS sets when $6\le m\le c_j\sqrt{n}$ for $c_j=1.3229/\sqrt{111+5j}$, 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 $A\subseteq D_{2n}$, proving an explicit formula for $|A-A|$ when $n$ is prime.
