In the infinite dihedral group every integer difference |AA^{-1}| - |A^{-1}A| is achievable while in the free group F_2 only even integers are achievable, with constructions and minimum sizes for nonzero difference depending on order-2 elements.
Many sets have more sums than differences
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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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SimpleTES scales test-time evaluation in LLMs to discover state-of-the-art solutions on 21 scientific problems across six domains, outperforming frontier models and optimization pipelines with examples like 2x faster LASSO and new Erdos constructions.
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Comparing Left and Right Quotient Sets in Groups
In the infinite dihedral group every integer difference |AA^{-1}| - |A^{-1}A| is achievable while in the free group F_2 only even integers are achievable, with constructions and minimum sizes for nonzero difference depending on order-2 elements.
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Evaluation-driven Scaling for Scientific Discovery
SimpleTES scales test-time evaluation in LLMs to discover state-of-the-art solutions on 21 scientific problems across six domains, outperforming frontier models and optimization pipelines with examples like 2x faster LASSO and new Erdos constructions.