Discrepancy

of a sequence $\omega=(\mathbf{x}_1,\ldots,\mathbf{x}_N)$ of points from the unit $s$-dimensional cube $K_s = \{\mathbf{x} : 0 \le x_\nu < 1\,,\ \nu=1,\ldots,s \}$

The norm of the functional \begin{equation}\label{eq:1} \phi(\alpha;\omega) = |V| - \frac{N(V)}{N}\,, \end{equation} calculated in some metric. Here, $|V|$ and $N(V)$ are, respectively, the volume of the domain $V = \{\mathbf{x} : 0 \le x_\nu < \alpha_\nu\,,\ \nu=1,\ldots,s \}$ and the number of the points of $\omega$ belonging to $V$. If one considers the distribution of the points of $\omega$ over domains of the type $V = \{\mathbf{x} : \alpha_\nu \le x_\nu < \beta_\nu\,,\ \nu=1,\ldots,s \}$, then, in formula (1), $\phi(\alpha;\omega)$ is usually replaced by $\phi(\alpha,\beta;\omega)$.

The following norms of the functional \eqref{eq:1} are most often used: $$ D_N(\omega) = \sup_{\alpha,\beta\in K_s} |\phi(\alpha,\beta;\omega)|\ , $$ $$ D_N^*(\omega) = \sup_{\alpha\in K_s} |\phi(\alpha;\omega)|\ , $$ $$ D_N(\omega,L_p) = \left({ \int_0^1\cdots\int_0^1 |\phi(\alpha;\omega)|^p d\alpha_1\ldots d\alpha_s }\right)^{1/p} \ . $$

A sequence $\omega=(\mathbf{x}_1,\ldots,\mathbf{x}_N,\ldots)$ of points from the $s$-dimensional unit cube $K_s$ is uniformly distributed if and only if [1] $$ \lim_{N\rightarrow\infty} D_N(\omega) = 0 \ . $$

For any infinite sequence $\omega=(x_1,\ldots,x_N,\ldots)$ of one-dimensional points the following theorem [3] is valid: $$ \limsup N D_N(\omega) = \infty \ . $$ For any such sequence $\omega$ it is possible to find a sequence $N_1,\ldots,N_k,\ldots$ such that for $N = N_k$ one has [4], $$ N D_N(\omega) > C_1 \sqrt{\log N} \ . $$ The final result [5] for infinite sequences of one-dimensional points is that for $N = N_k$: $$ N D_N(\omega) > C_2 \log N \ . $$

Studies were made of the discrepancies of various concrete sequences [6]–[8], and the estimates from above $$ N D_N(\omega,L_2) \le C_3(s) \log^{s+1} N \ , $$ $$ N D_N(\omega) \le C_4(s) \log^s N $$ were obtained, respectively, for finite and infinite sequences, as well as an estimate from below [4]: For any sequence of $N$ points, the following inequality is valid: $$ N D_N(\omega,L_2) \ge C_5(s) \log^{(s+1)/2} N \ . $$

For any infinite sequence $\omega = \{\mathbf{x}_n \in K_s \}$ it is possible to find a sequence of numbers $N_1,\ldots,N_k,\ldots$ such that for $N = N_k$ one has $$ N D_N(\omega,L_2) \ge C_6(s) \log^{s/2} N \ . $$

Also, $$ D_N(\omega) \ge D_N(\omega,L_2) \ . $$

Comments

See also Distribution modulo one; Distribution modulo one, higher-dimensional; Uniform distribution.

References

[1]	H. Weyl, "Ueber die Gleichverteilung von Zahlen mod Eins" Math. Ann. , 77 (1916) pp. 313–352
[2]	J.G. van der Corput, "Verteilungsfunktionen" Proc. Koninkl. Ned. Akad. Wet. A , 38 : 8 (1935) pp. 813–821; 1058–1066
[3]	T. van Aardenne-Ehrenfest, "On the impossibility of a just distribution" Indag. Math. , 11 (1949) pp. 264–269
[4]	K.F. Roth, "On irregularities of distribution" Mathematika , 1 (1954) pp. 73–79 Zbl 0057.28604
[5]	W.M. Schmidt, "Irregularities of distribution VII" Acta Arithm. , 21 (1972) pp. 45–50
[6]	J.H. Halton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals" Numer. Math. , 2 : 2 (1960) pp. 84–90
[7]	I.M. Sobol', "The distribution of points in a cube and the approximate evaluation of integrals" USSR Comp. Math. and Math. Phys. , 7 : 4 (1967) pp. 86–112 Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 4 (1967) pp. 784–802
[8]	N.M. Korobov, "Number-theoretical methods in approximate analysis" , Moscow (1963) (In Russian)
[9]	L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974) Zbl 0281.10001; repr. Dover (2006) ISBN 0-486-45019-8
[a1]	J. Beck, W.L. Chen, "Irregularities of distribution" , Cambridge Univ. Press (1987) Zbl 0617.10039