# Small sumsets in real numbers : a continuous 3k-4 theorem

Abstract : We prove a continuous Freiman's 3k-4 theorem for small sumsets in R by using some ideas from Ruzsa's work on measure of sumsets in R as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when $\lambda(A+B)<\lambda(A)+\lambda(B)+\min(\lambda(A),\lambda(B))$. We also give some structural information for sets of large density with small sumset and characterize the extremal sets for which equality holds in the lower bounds for $\lambda(A+B)$.
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Cited literature [11 references]

https://hal.archives-ouvertes.fr/hal-01316201
Contributor : Anne de Roton <>
Submitted on : Sunday, May 15, 2016 - 9:07:25 PM
Last modification on : Wednesday, October 10, 2018 - 1:08:44 PM
Long-term archiving on : Wednesday, November 16, 2016 - 5:49:12 AM

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• HAL Id : hal-01316201, version 1

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Anne de Roton. Small sumsets in real numbers : a continuous 3k-4 theorem. 2016. ⟨hal-01316201⟩

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