论文标题
较高维度的广义MSTD集的构造
Constructions of Generalized MSTD Sets in Higher Dimensions
论文作者
论文摘要
令$ a $为一组有限的整数，定义$$ a+a+a \ = \ {a_1+a_2：a_1，a_1，a_2 \ in a \}，\ \ \ \ \ \ \ \ \ a-a \ = \ \ \ \ \ {a_1-a_2：a_1，a_2：a_1，a_2 \ in a_1，a_2 \ in a \ $ and $ and $ and $ and $ and $ and $ and $ and $ shote and $ shote and $ shote and $ nont和sn $$ sa-da \ = \ \ sustembrace {a+\ cdots+a} _ {s} - \ underbrace {a- \ cdots-a} _ {d}。 > | a-a | $。最初认为，MSTD的$ [0，n] $的子集的百分比将归零，因为$ n $接近无穷大，因为加法是交换性的，而减去则不是。但是，在2006年令人惊讶的结果中，马丁和奥布莱恩特证明了一个正百分比为MSTD，尽管该百分比非常小，大约$ 10^{-4} $％。 This result was extended by Iyer, Lazarev, Miller, ans Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for $\{s_1,d_1\} \neq \{s_2, d_2\}$ and $s_1+d_1=s_2+d_2$ with $|s_1A-d_1A| > | s_2a-d_2a | $，在$ d $ dimensions中，正组为MSTD。 对于许多这样的结果，建立明确的MSTD套件以$ 1 $的限制依赖于该集合的左右边缘的特定选择，以迫使某些差异在达到所需的款项。在较高的维度中，几何形状更仔细地评估了哪些元素的行为与$ 1 $维的边缘元素相同。我们在$ d $ dimensions中学习条纹，并使用它们来创建新的显式结构。我们证明存在$ d $ dimensions中的广义MSTD集和$ k $ - 代理集的存在，这些集合是所有$ 1 \ leq c \ leq c \ leq k $的$ | ca+ca+ca+ca |> | ca-ca | $。然后，我们证明在某些条件下，没有$ | ka+ka |> | ka-ka | $ in \ mathbb {n}。
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 $$sA-dA\ =\ \underbrace{A+\cdots+A}_{s} -\underbrace{A-\cdots-A}_{d}.$$ 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, ans Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for $\{s_1,d_1\} \neq \{s_2, d_2\}$ and $s_1+d_1=s_2+d_2$ with $|s_1A-d_1A| > |s_2A-d_2A|$, 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 $1\leq c \leq k$. We then prove that under certain conditions, there are no sets with $|kA+kA|>|kA-kA|$ for all $k \in \mathbb{N}.$
