In the branch of mathematics known as additive combinatorics, Kneser's theorem can refer to one of several related theorems regarding the sizes of certain sumsets in abelian groups.
These are named after Martin Kneser, who published them in 1953[1] and 1956.
[2] They may be regarded as extensions of the Cauchy–Davenport theorem, which also concerns sumsets in groups but is restricted to groups whose order is a prime number.
[3] The first three statements deal with sumsets whose size (in various senses) is strictly smaller than the sum of the size of the summands.
The last statement deals with the case of equality for Haar measure in connected compact abelian groups.
is an abelian group and
are nonempty finite subsets of
This statement is a corollary of the statement for LCA groups below, obtained by specializing to the case where the ambient group is discrete.
A self-contained proof is provided in Nathanson's textbook.
[4] The main result of Kneser's 1953 article[1] is a variant of Mann's theorem on Schnirelmann density.
, the lower asymptotic density of
lim inf
Kneser's theorem for lower asymptotic density states that if
, then there is a natural number
satisfies the following two conditions: and Note that
be an LCA group with Haar measure
denote the inner measure induced by
is Hausdorff, as usual).
We are forced to consider inner Haar measure, as the sumset of two
-measurable sets can fail to be
Satz 1 of Kneser's 1956 article[2] can be stated as follows: If
is compact and open.
is compact and open (and therefore
-measurable), being a union of finitely many cosets of
Because connected groups have no proper open subgroups, the preceding statement immediately implies that if
Examples where can be found when
Satz 2 of Kneser's 1956 article[2] says that all examples of sets satisfying equation (1) with non-null summands are obvious modifications of these.
is a connected compact abelian group with Haar measure
, and equation (1), then there is a continuous surjective homomorphism
and there are closed intervals