A set A is MSTD (more-sum-than-di?erence) if |A + A| > |A A|. Though MSTD sets are rare, Martin and O’Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r ! 1. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r ! 1. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an ecient method to partition {1, 2, . . . , r} (for r suciently large) into k 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sucient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r ! 1.
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机译：一个集合A是MSTD（更多 - 总和而不是-i？erence）IF | A + A | > | A a |。虽然MSTD集是罕见的，但Martin和O'Bryant证明了{1,2的MSTD子集的比例存在正常常数下限。 。 。 ，r}作为r！ 1.后来，亚拉达等。表明，{1,2的分解比例存在正常常数下限。 。 。 ，r}分为两个mstd子集作为r！但是，该方法是概率性的，并且没有发出显式分解。继续这项工作，我们提供了一个恩典的方法来分区{1,2，。 。 。 ，R}（对于r sugiently大）进入K 2 MSTD子集，积极回答Asada等人提出的问题。关于这是否可以所有这些k。接下来，让R（k）是最小的整数，使得所有R r（k），{1,2，。 。 。 ，R}可以将K-分解成MSTD子集。我们为r（k）建立粗糙的下限和上限。最后，当{1,2的分解比例存在正常恒定的下限时，我们提供了一个成功的条件。 。 。 ，r}进入K MSTD子集作为r！ 1。
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