Infinite Families of Partitions into MSTD Subsets

Abstract

A set $A$ is MSTD (more-sum-than-difference) 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 $\{1,2,\ldots ,r\}$ as $r\rightarrow\infty$. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of $\{1,2,\ldots,r\}$ into two MSTD subsets as $r\rightarrow\infty$. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition $\{1,2,\ldots,r\}$ (for $r$ sufficiently large) into $k \ge 2$ 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 $r\ge R(k)$, $\{1,2,\ldots,r\}$ can be $k$-decomposed into MSTD subsets. We establish rough lower and upper bounds for $R(k)$. Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of $\{1,2,\ldots,r\}$ into $k$ MSTD subsets as $r\rightarrow \infty$.