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
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The density of the sequence is the quantity
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The arithmetic sum of two sequences and
is the sequence
consisting of all possible sums
, where
and
. Mann's theorem asserts that
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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) |
Mann theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mann_theorem&oldid=15059