Infinite Families of Partitions into MSTD Subsets

TL;DR

This paper develops an efficient method to partition large initial segments of natural numbers into multiple MSTD subsets, addressing open questions about explicit decompositions and bounds on the minimal segment size for such partitions.

Contribution

It introduces a constructive approach for partitioning sets into multiple MSTD subsets and establishes bounds on the minimal size needed for these partitions.

Findings

Provided an explicit method for partitioning into k MSTD subsets for large r.

Established bounds for the minimal r (R(k)) for such partitions to exist.

Identified conditions for a positive proportion of such partitions as r grows.

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 $r\rightarrow \infty$.