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On the Schnirelmann density of sumsets

A classical result of Schnirelmann asserts that σ(A + B) ≥ σ(A) + σ(B)−σ(A)σ(B) for arbitrary sets of integers, provided at least one of them contains 0. We establish the best possible bound that can be asserted in this case, in particular we describe all cases of equality in Schinrelmann’s theorem.