Many sets have more sums than differences
Number Theory
2010-03-04 v3 Combinatorics
Abstract
Since addition is commutative but subtraction is not, the sumset S+S of a finite set S is predisposed to be smaller than the difference set S-S. In this paper, however, we show that each of the three possibilities (|S+S|>|S-S|, |S+S|=|S-S|, |S+S|<|S-S|) occur for a positive proportion of the subsets of {0, 1, ..., n-1}. We also show that the difference |S+S| - |S-S| can take any integer value, and we show that the expected number of omitted differences is asymptotically 6 while the expected number of missing sums is asymptotically 10. Other data and conjectures on the distribution of these quantities are also given.
Cite
@article{arxiv.math/0608131,
title = {Many sets have more sums than differences},
author = {Greg Martin and Kevin O'Bryant},
journal= {arXiv preprint arXiv:math/0608131},
year = {2010}
}
Comments
20 pages, 5 figures; this version contains many theorems and conjectures not in earlier versions