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

Anne de Roton (1)

1 IECL - Institut Élie Cartan de Lorraine

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 $\lambda(A+B)$.

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

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