Constructions of generalized MSTD sets in higher dimensions
Constructions of generalized MSTD sets in higher dimensions
复制标题
高维广义 MSTD 集的构造
DOI：
10.1016/j.jnt.2021.03.029
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发表时间：
2022
影响因子：
0.7
通讯作者：
Miller, Steven J.
中科院分区：
文献类型：
--
作者：
Kim, Elena;Miller, Steven J.
设A是一组有限整数，定义A+ A={a 1+ a 2：a 1，a 2∈ A}，A− A={a 1− a 2：a 1，a 2∈ A}，对于非负整数s和d定义s A− d A= A+ n + A s− A− n − A d。和大于差（MSTD）集是A，其中|A+ A|>| A− A|.最初认为，当n接近无穷大时，[0，n]的子集中MSTD的百分比将变为零，因为加法是可交换的，而减法不是。然而，在2006年的一个令人惊讶的结果中，马丁和奥布莱恩特证明了一个积极的百分比是MSTD，尽管这个百分比非常小，大约是10 - 4%。Iyer，Lazarev，米勒和Zhang [ILMZ]扩展了这一结果，他们证明了正百分比的集合是广义MSTD集合，{s1，d1}的集合≠{s2，d2}且s1 + d1 = s2 + d2，|s 1 A− d 1 A|>| s 2 A− d 2 A|，并且在d维中，正百分比的集合是MSTD。对于许多这样的结果，在1维中建立显式MSTD集合依赖于集合的左边缘和右边缘上的元素的特定选择，以迫使某些差异被遗漏，同时获得期望的和。在更高的维度中，几何形状迫使更仔细地评估哪些元素具有与一维边缘元素相同的行为。我们研究条纹在d维，并使用这些来创建新的明确的建设。我们证明了d维广义MSTD集的存在性和k代集的存在性，这些集是其中|c A+ c A|>| c A− c A|对于所有1≤ c≤ k。然后，我们证明，在一定的条件下，没有集合，|k A+ k A|>| k A− k A|对于所有k∈ N。视频有关本文的视频摘要，请访问https：//youtu。是/rojbhVqN 90 Q。
Text Let A be a set of finite integers, define A+ A={a 1+ a 2: a 1, a 2∈ A}, A− A={a 1− a 2: a 1, a 2∈ A}, and for non-negative integers s and d define s A− d A= A+⋯+ A︸ s− A−⋯− A︸ d. A More Sums than Differences (MSTD) set is an A where| A+ A|>| A− A|. It was initially thought that the percentage of subsets of [0, n] that are MSTD would go to zero as n approaches infinity as addition is commutative and subtraction is not. However, in a surprising 2006 result, Martin and O'Bryant proved that a positive percentage of sets are MSTD, although this percentage is extremely small, about 10− 4 percent. This result was extended by Iyer, Lazarev, Miller, and Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for {s 1, d 1}≠{s 2, d 2} and s 1+ d 1= s 2+ d 2 with| s 1 A− d 1 A|>| s 2 A− d 2 A|, and that in d-dimensions, a positive percentage of sets are MSTD. For many such results, establishing explicit MSTD sets in 1-dimensions relies on the specific choice of the elements on the left and right fringes of the set to force certain differences to be missed while desired sums are attained. In higher dimensions, the geometry forces a more careful assessment of what elements have the same behavior as 1-dimensional fringe elements. We study fringes in d-dimensions and use these to create new explicit constructions. We prove the existence of generalized MSTD sets in d-dimensions and the existence of k-generational sets, which are sets where| c A+ c A|>| c A− c A| for all 1≤ c≤ k. We then prove that under certain conditions, there are no sets with| k A+ k A|>| k A− k A| for all k∈ N. Video For a video summary of this paper, please visit https://youtu. be/rojbhVqN90Q.