Fringe pairs in generalized MSTD sets

Abstract

A More Sums Than Differences (MSTD) set is a set $A$ for which $|A+A|>|A-A|$. Martin and O'Bryant proved that the proportion of MSTD sets in $\{0,1,\dots ,n\}$ is bounded below by a positive number as $n$ goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set $A$ for which $|sA-dA|>|\sigma A-\delta A|$ for a prescribed $s+d=\sigma +\delta$. We offer efficient constructions of $k$-generational MSTD sets, sets $A$ where $A,A+A,\dots ,kA$ are all MSTD. We also offer an alternative proof that the proportion of sets $A$ for which $|sA-dA|-|\sigma A-\delta A|=x$ is positive, for any $x\in \mathbb{Z}$. We prove that for any $\epsilon >0$, $\mathrm{Pr}(1-\epsilon <\log |sA-dA|/\log |\sigma A-\delta A|<1+\epsilon )$ goes to $1$ as the size of $A$ goes to infinity and we give a set $A$ which has the current highest value of $\log |A+A|/\log |A-A|$. We also study decompositions of intervals $\{0,1,\dots ,n\}$ into MSTD sets and prove that a positive proportion of decompositions into two sets have the property that both sets are MSTD.

Version 2.0 (23 pages)