Small sumsets in $\protect \mathbb{R}$: full continuous $3k-4$ theorem, critical sets
Résumé
We prove a full 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 λ(A + B) < λ(A) + 2λ(B) and either λ(A) ≥ λ(B) or A has larger diameter than B. We also give some structural information for sets of large density according to the size of their sumset, a result so far unknown in the discrete and the continuous setting. Finally, we characterize the critical sets for which equality holds in the lower bounds for λ(A + B).
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