additive number theory Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigr ...

, Kemnitz's conjecture states that every set of

lattice point In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...

s in the plane has a large subset whose

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows: :Let

n

be a natural number and

S

a set of

4n-3

lattice points in plane. Then there exists a subset

S_1 \subseteq S

with

n

points such that the centroid of all points from

S_1

is also a lattice point. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the

Erdős–Ginzburg–Ziv theorem In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group ''G'' and a positive integer ''n'', one asks for the smallest value of ''k'' suc ...

, an analogous one-dimensional result stating that every

2n-1

integers have a subset of size

n

whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with

4n-2

lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.

References

Further reading