Mann theorem

A theorem giving an estimate of the density of the sum of two sequences (cf. Density of a sequence). Let be an increasing sequence of integers and let

The density of the sequence is the quantity

The arithmetic sum of two sequences and is the sequence consisting of all possible sums , where and . Mann's theorem asserts that

Mann's theorem implies that if is a sequence of positive density less than 1 and is another sequence of positive density, then on addition of and the density is increased. Another important consequence of Mann's theorem is: Each sequence of positive density is a basis for the sequence of natural numbers. Mann's theorem essentially strengthens a similar theorem of Shnirel'man (cf. Shnirel'man method). It was proved by H.B. Mann [1].

References

[1]	H.B. Mann, "A proof of the fundamental theorem on the density of sums of sets of positive integers" Ann. of Math. , 43 (1942) pp. 523–527
[2]	H.H. Ostmann, "Additive Zahlentheorie" , Springer (1956)
[3]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian)