Discrepancy

of a sequence of points from the unit -dimensional cube

The norm of the functional

(1)

calculated in some metric. Here, and are, respectively, the volume of the domain and the number of the points of belonging to . If one considers the distribution of the points of over domains of the type , then, in formula (1), is usually replaced by .

The following norms of the functional (1) are most often used:

A sequence of points from the -dimensional unit cube is uniformly distributed if and only if [1]

For any infinite sequence of one-dimensional points the following theorem [3] is valid:

For any sequence it is possible to find a sequence such that for one has [4],

The final result [5] for infinite sequences of one-dimensional points is that for :

Studies were made of the discrepancies of various concrete sequences [6]–[8], and the estimates from above

were obtained, respectively, for finite and infinite sequences, as well as an estimate from below [4]: For any sequence of points, the following inequality is valid:

For any infinite sequence it is possible to find a sequence of numbers such that for one has

Also,

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
[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)

Comments

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

References

[a1]	J. Beck, W.L. Chen, "Irregularities of distribution" , Cambridge Univ. Press (1987)