PlanetMath: Schnirelmann density

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Schnirelmann density

(Definition)

Let be a subset of $\mathbb{Z}$ , and let be number of elements of in . Schnirelmann density of is

$\displaystyle \sigma A = \inf_n \frac{A(n)}{n}.$

Schnirelmann density has the following properties:

Schnirelmann proved that if $0 \in A \cap B$ then

$\displaystyle \sigma(A+B)\geq \sigma A + \sigma B - \sigma A \cdot \sigma B$

and also if $\sigma A + \sigma B \geq 1$ , then $\sigma (A+B)=1$ . From these he deduced that if $\sigma A>0$ then

"Schnirelmann density" is owned by bbukh.

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Other names:

Shnirel'man density, Shnirelman density

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Cross-references: additive basis, properties, number, subset
There are 3 references to this entry.

This is version 6 of Schnirelmann density, born on 2002-12-26, modified 2006-09-05.
Object id is 3838, canonical name is SchnirlemannDensity.
Accessed 2795 times total.

Classification:

AMS MSC:	11B05 (Number theory :: Sequences and sets :: Density, gaps, topology)
	11B13 (Number theory :: Sequences and sets :: Additive bases)

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