Distribution of Missing Sums in Sumsets
Abstract
For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of A+A given the uniform distribution on subsets A of {0, 1, ..., n1}. They also conjectured the existence of a limiting distribution for A+A and showed that the expectation of A+A is 2n  11 + O((3/4)^{n/2}). Zhao proved that the limits m(k) := lim_{n > oo} Prob(2n1A+A=k) exist, and that sum_{k >= 0} m(k)=1. We continue this program and give exponentially decaying upper and lower bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the distribution is at least bimodal; sumsets have an unexpected bias against missing exactly 7 sums. The proof of the latter is by reduction to questions on the distribution of related random variables, with large scale numerical computations a key ingredient in the analysis. We also derive an explicit formula for the variance of A+A in terms of Fibonacci numbers, finding Var(A+A) is approximately 35.9658. New difficulties arise in the form of weak dependence between events of the form {x in A+A}, {y in A+A}. We surmount these obstructions by translating the problem to graph theory. This approach also yields good bounds on the probability for A+A missing a consecutive block of length k.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.4700
 Bibcode:
 2011arXiv1109.4700L
 Keywords:

 Mathematics  Number Theory;
 11P99 (Primary) 11K99 (Secondary)
 EPrint:
 To appear in Experimental Mathematics. Version 3. Larger computations than before, conclusively proving the divot exists. 40 pages, 15 figures