Distribution modulo one, higher-dimensional

The distribution of the fractional parts of a sequence of elements of the -dimensional Euclidean space in the -dimensional unit cube . Here denotes the fractional part of a number.

The sequence , is said to be uniformly distributed in if the equality

holds for any rectangle , where is the number of those points among the first members of the sequence which belong to and is the measure of .

A sequence , is said to be uniformly distributed modulo one if the corresponding sequence of fractional parts is uniformly distributed in .

Weyl's criterion for higher-dimensional distribution modulo one.

A sequence , is uniformly distributed in if and only if

for any set of integers . A particular case of this theorem is the Weyl criterion for a sequence of real numbers to be uniformly distributed modulo one. Weyl's criterion implies the following theorem of Kronecker: Let be real numbers that are linearly independent over the field of rational numbers, let be arbitrary real numbers and let and be positive numbers; then there are integers and such that

for all . In other words, the sequence , is uniformly distributed modulo one.

References

[1]	J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)

Comments

For additional references see Distribution modulo one.