Schnirelmann density

The Schnirelmann density is sensitive to the first values of a set:

Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis.

Asymptotic density contrasts, for example, with the Schnirelmann density.

Consequently, the Schnirelmann densities of the even numbers and the odd numbers, which one might expect to agree, are 0 and 1/2 respectively.

An elementary proof of the Hilbert-Waring theorem; see also Schnirelmann density.

In fact, we still have , and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase.

In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is.

The Schnirelmann density is well-defined even if the limit of A(n)/n as n fails to exist (see asymptotic density).

Khintchin proved that the sequence of squares, though of zero Schnirelmann density, when added to a sequence of Schnirelmann density between 0 and 1, increases the density:

It follows from the fact that sum-free sequences are small that they have zero Schnirelmann density; that is, if is defined to be the number of sequence elements that are less than or equal to , then .