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Let A be a set of finite integers, define$A+A=\{a_1+a_2:a_1,a_2\in A\},A-A=\{a_1-a_2:a_1,a_2\in A\},$ and for non-negative integers s and d define$sA-dA=\underset{s}{\underbrace{A+\cdots +A}}-\underset{d}{\underbrace{A-\cdots -A}}.$ 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 $\{s_1,d_1\}\ne \{s_2,d_2\}$ 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 $|cA+cA|>|cA-cA|$ for all $1\le c\le k$. We then prove that under certain conditions, there are no sets with $|kA+kA|>|kA-kA|$ for all $k\in \mathbb{N}$.

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中文翻译：

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高维广义 MSTD 集的构造

文本

令A为一组有限整数，定义$\mathrm{一种}+\mathrm{一种}=\{{\mathrm{一种}}_1+{\mathrm{一种}}_2：{\mathrm{一种}}_1,{\mathrm{一种}}_2\in \mathrm{一种}\},\mathrm{一种}-\mathrm{一种}=\{{\mathrm{一种}}_1-{\mathrm{一种}}_2：{\mathrm{一种}}_1,{\mathrm{一种}}_2\in \mathrm{一种}\},$并且对于非负整数s和d定义$s\mathrm{一种}-d\mathrm{一种}=\underset{s}{\underbrace{\mathrm{一种}+\cdots +\mathrm{一种}}}-\underset{d}{\underbrace{\mathrm{一种}-\cdots -\mathrm{一种}}}.$A More Sums than Differences (MSTD) 集是A，其中$|\mathrm{一种}+\mathrm{一种}|>|\mathrm{一种}-\mathrm{一种}|$. 最初认为子集的百分比$[0,n]$当n接近无穷大时，MSTD 将变为零，因为加法是可交换的，而减法不是。然而，在 2006 年令人惊讶的结果中，Martin 和 O'Bryant 证明了 MSTD 的正百分比，尽管这个百分比非常小，大约${10}^{-4}$百分。Iyer、Lazarev、Miller 和 Zhang [ILMZ] 扩展了这一结果，他们表明正百分比的集合是广义 MSTD 集合，集合为$\{s_1,d_1\}\ne \{s_2,d_2\}$和$s_1+d_1=s_2+d_2$和$|s_1\mathrm{一种}-d_1\mathrm{一种}|>|s_2\mathrm{一种}-d_2\mathrm{一种}|$，并且在d维中，正百分比的集合是 MSTD。

对于许多这样的结果，在一维中建立显式 MSTD 集依赖于对集左右边缘元素的特定选择，以在获得所需总和的同时强制忽略某些差异。在更高维度中，几何结构迫使对哪些元素具有与一维边缘元素相同的行为进行更仔细的评估。我们研究d维中的条纹，并使用它们来创建新的显式结构。我们证明了d维中广义 MSTD 集的存在和k代集的存在，这些集是$|C\mathrm{一种}+C\mathrm{一种}|>|C\mathrm{一种}-C\mathrm{一种}|$对所有人$1\le C\le ķ$. 然后我们证明在特定条件下，不存在具有$|ķ\mathrm{一种}+ķ\mathrm{一种}|>|ķ\mathrm{一种}-ķ\mathrm{一种}|$对所有人$ķ\in \mathbb{ñ}$.

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