Distribution modulo one, higher-dimensional

The distribution of the fractional parts $\{P_j\} = (\{x_1^{(j)}\},\ldots,\{x_n^{(j)}\})$ of a sequence $P_j = (x_1^{(j)},\ldots,x_n^{(j)})$ of elements of the $n$-dimensional Euclidean space in the $n$-dimensional unit cube $E = \{ (x_1,\ldots,x_n) : 0 \le x_i < 1\,\ i=1,\ldots n\}$. Here $\{x\}$ denotes the fractional part of a number $x$.

The sequence $\{P_j\}$, $j=1,2,\ldots$ is said to be uniformly distributed in $E$ if the equality $$ \lim_{m\rightarrow\infty} \frac{\phi_m(V)}{m} = |V| $$ holds for any rectangle $V$, where $\phi_m(V)$ is the number of those points among the first $m$ members of the sequence which belong to $V$ and $|V|$ is the measure of $V$.

A sequence $P_j$, $j=1,2,\ldots$ is said to be uniformly distributed modulo one if the corresponding sequence of fractional parts $\{P_j\}$ is uniformly distributed in $E$.

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

A sequence $P_j$, $j=1,2,\ldots$ is uniformly distributed in $E$ if and only if $$ \lim_{N\rightarrow\infty} \frac{1}{N} \sum_{j=1}^N \exp\left({ 2\pi i (a_1x_1^{(j)}+\cdots+a_n x_n^{(j)}) }\right) = 0 $$ for any set of integers $(a_1,\ldots,a_n) \ne (0,\ldots,0)$. 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 $\theta_1,\ldots,\theta_n,1$ be real numbers that are linearly independent over the field of rational numbers, let $\alpha_1,\ldots,\alpha_n$ be arbitrary real numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that $$ m>N\,,\ \ \ |m\theta_i-p_i-\alpha_i| < \epsilon $$ for all $i=1,\ldots,n$. Indeed, the sequence $m\theta = (m\theta_1,\ldots,m\theta_n)$, $m=1,2,\ldots$ 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.