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Many sets have more sums than differences

Greg Martin·Kevin O'Bryant
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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.

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Greg Martin, Kevin O'Bryant. 2006-12-04. Many sets have more sums than differences. https://arxiv.org/abs/math/0608131

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