Let 
,
,
... be an infinite series of real numbers lying between 0 and 1. Then corresponding
to any arbitrarily large 
, there exists a positive integer 
and two subintervals of equal length such that the number
of 
with 
,
2, ..., 
which lie in one of the subintervals differs from the number of such 
that lie in the other subinterval by more than 
(van der Corput 1935ab, van Aardenne-Ehrenfest 1945, 1949,
Roth 1954).
This statement can be refined as follows. Let 
be a large integer and 
, 
, ..., 
be a sequence of 
real numbers lying between 0 and 1. Then for any integer 
and any real number 
satisfying 
, let 
denote the number of 
with 
, 2, ..., 
that satisfy 
. Then there exist 
and 
such that
 |
(1)
|
where 
is a positive constant. Schmidt (1972) improved upon this result to obtain
 |
(2)
|
which is essentially optimal.
This result can be further strengthened, which is most easily done by reformulating the problem. Let 
be an integer and 
,
,
..., 
be 
(not necessarily distinct) points in the square 
, 
. Then
![int_0^1int_0^1[S(x,y)-Nxy]^2dxdy>c_2lnN,](/images/equations/DiscrepancyTheorem/NumberedEquation3.svg) |
(3)
|
where 
is a positive constant and 
is the number of points in the rectangle 
, 
(Roth 1954). Therefore,
 |
(4)
|
and the original result can be stated as the fact that there exist 
and 
such that
 |
(5)
|
The randomly distributed points shown in the above squares have 
and 9.11, respectively.
Similarly, the discrepancy of a set of 
points in a unit 
-hypercube satisfies
 |
(6)
|
(Roth 1954, 1976, 1979, 1980).
