Integer sets with doubling at most 4 + δ for sufficiently small δ > 0 have a specific arithmetic structure, generalizing the doubling < 4 case.
Raghavan,Improved Bounds for the Freiman-Ruzsa Theorem
2 Pith papers cite this work. Polarity classification is still indexing.
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Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.
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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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Entropy lower bounds and sum-product phenomena
Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.