Mann theorem

2020 Mathematics Subject Classification: Primary: 11B05 [MSN][ZBL]

A theorem giving an estimate of the density of the sum of two sequences (cf. Density of a sequence), proved by H.B. Mann [1]. Let $A = (0 < a_1 < a_2 < \cdots)$ be an increasing sequence of integers and let $$ A(n) = \sum_{a_i \le n \\ a_i \in A} 1 $$ be the counting function of $A$. The density of the sequence $A$ is the quantity $$ d(A) = \inf_n \frac{A(n)}{n} \ . $$

The arithmetic sum of two sequences $A$ and $B$ is the sequence $C = A+B$ consisting of all possible sums $c=a+b$, where $a \in A$ and $b \in B$. Mann's theorem asserts that $$ d(A+B) \ge \min\{d(A)+d(B),1\} \ . $$

Mann's theorem implies that if $A$ is a sequence of positive density less than 1 and $B$ is another sequence of positive density, then on addition of $B$ to $A$ 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), that $d(A+B) \ge d(A) + d(B) - d(A)d(B)$.

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 Zbl 0061.07406
[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)