Generalized More Sums Than Differences sets

Abstract

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A ⊂ Z such that |A + A| < |A − A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant [MO] proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that |ǫ 1 A + · · · + ǫ k A| > |δ 1 A + · · · + δ k A| a positive percent of the time for all nontrivial choices of ǫ j , δ j ∈ {−1, 1}. Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets.We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken. For example, we prove that for any m, |ǫ 1 A+· · ·+ǫ k A|−|δ 1 A+· · ·+δ k A| = m a positive percentage of the time. We find the limiting behavior of kA = A + · · · + A for an arbitrary set A as k → ∞ and an upper bound of k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if A, A + A, . . . , kA are all sum-dominant. Numerical searches were unable to find even a 2-generational set (heuristics indicate that the probability is at most 10 −9 , and quite likely significantly less). We prove that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k.

Fringe pairs in generalized MSTD sets

“…• |A| is the cardinality of A, Our first result significantly improves upon the constructions of k-generational sets given by Iyer et al [4], which used base expansion and therefore yielded sets A which grow astronomically with k. Theorem 1.1. For any q > 2, there exists a set A with |A| = O(q 2 ) such that for all s + d = σ + δ with s > σ, |sA − dA| > |σA − δA|.…”

“…However, a major drawback of base expansion is that the set C grows large very quickly (we explore this issue in greater detail in §5, where we investigate the ratio of the logarithms of the cardinalities). According to the construction in [4], the middle of A j has at least 2(2jr − 4j + 1) elements where r = 4j + 2. Thus |A j | = Ω(j 2 ), which means there is a constant c such that…”

“…Some progress has been made in constructing infinite families of such sets [2,6,7,[9][10][11][12][13][14][15], as well as MSTD sets with additional properties. One example is due to Iyer, Lazarev, Miller and Zhang [4], who developed constructions for generalized MSTD sets, sets A that satisfy |sA − dA| > |σA − δA| for a given s + d = σ + δ. This provided them with the framework to construct k-generational sets, a rarer class of MSTD sets A for which A, A + A, .…”

“…Our first set of results concern fringe constructions for generalized MSTD sets that are much more efficient than the ones in [4]. Before stating these we first set some notation:…”

The Haight–Ruzsa Method for Sets with More Differences than Multiple Sums

“…A Several families of MSTD sets of integers have been constructed, but there is no classification of such sets and many unsolved problems remain (cf. Hegarty [3], Hegarty and Miller [2], Iyer, Lazarov, Miller, and Zhang [4,5], Martin and O'Bryant [6], Nathanson [7,8]). A dual problem is to construct sets with more differences than multiple sums, that is, finite sets A in a group W such that the difference set |A − A| is large but the h-fold sumset hA is small.…”