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Difference between revisions of "Mann theorem"

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A theorem giving an estimate of the density of the sum of two sequences (cf. [[Density of a sequence|Density of a sequence]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622301.png" /> be an increasing sequence of integers and let
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{{TEX|done}}{{MSC|11B05}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622302.png" /></td> </tr></table>
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A theorem giving an estimate of the density of the sum of two sequences (cf. [[Density of a sequence]]), proved by H.B. Mann [[#References|[1]]]. Let $A = (0 < a_1 < a_2 < \cdots)$ be an increasing sequence of integers and let
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$$
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A(n) = \sum_{a_i \le n \\ a_i \in A} 1
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$$ be the counting function of $A$.  The density of the sequence $A$ is the quantity
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$$
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d(A) = \inf_n \frac{A(n)}{n} \ .
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$$
  
The density of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622303.png" /> is the quantity
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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
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$$
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d(A+B) \ge \min\{d(A)+d(B),1\} \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622304.png" /></td> </tr></table>
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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 $d(A+B) \ge d(A) + d(B) - d(A)d(B)$.
 
 
The arithmetic sum of two sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622306.png" /> is the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622307.png" /> consisting of all possible sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622308.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m0622309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223010.png" />. Mann's theorem asserts that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223011.png" /></td> </tr></table>
 
 
 
Mann's theorem implies that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223012.png" /> is a sequence of positive density less than 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223013.png" /> is another sequence of positive density, then on addition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062230/m06223015.png" /> the density is increased. Another important consequence of Mann's theorem is: Each sequence of positive density is a [[Basis|basis]] for the sequence of natural numbers. Mann's theorem essentially strengthens a similar theorem of Shnirel'man (cf. [[Shnirel'man method|Shnirel'man method]]). It was proved by H.B. Mann [[#References|[1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Ostmann,  "Additive Zahlentheorie" , Springer  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. Gel'fond,  Yu.V. Linnik,  "Elementary methods in the analytic theory of numbers" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  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}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Ostmann,  "Additive Zahlentheorie" , Springer  (1956)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. Gel'fond,  Yu.V. Linnik,  "Elementary methods in the analytic theory of numbers" , M.I.T.  (1966)  (Translated from Russian)</TD></TR>
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</table>

Latest revision as of 11:41, 19 November 2017

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)
How to Cite This Entry:
Mann theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mann_theorem&oldid=15059
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article