Integer sets with doubling at most 4 + δ for sufficiently small δ > 0 have a specific arithmetic structure, generalizing the doubling < 4 case.
Eberhard,The abelian arithmetic regularity lemma
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For dense subsets A of F_p with |A| ≥ αp, max(|A+A|, |A·A|) ≥ (f(α) - o(1))p where f(α) is the optimal constant explicitly determined here.
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A structure theorem for sets with doubling $4+\delta$
Integer sets with doubling at most 4 + δ for sufficiently small δ > 0 have a specific arithmetic structure, generalizing the doubling < 4 case.
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The sum-product phenomenon for dense subsets of finite fields
For dense subsets A of F_p with |A| ≥ αp, max(|A+A|, |A·A|) ≥ (f(α) - o(1))p where f(α) is the optimal constant explicitly determined here.