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Probabilistic methods prove existence results in a nonconstructive fashion, by showing the chance of randomly selecting a solution is greater than zero.
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Does $|A+A|$ concentrate near its mean?
If $\epsilon>1/6$ then $|A+A|=N$ with high probability. Suppose that $\epsilon<1/6$. Call a pair $(x,y) \in {\mathbb Z}/N{\mathbb Z} \times {\mathbb Z}/N{\mathbb Z} $ "bad" if $(x,y) \in A \times A $ …