On Irregularities of Distribution
Nan Zhao
Reference
[1] K. F. Roth, "On irregularities of distribution," Mathematika, 1: 73-79,
1954.
[2] J. Beck, “A two-dimentional van Aardenne-Ehrenfest theorem in
irregularities of distribution”, Compositio Mathematica, 72: 269-339, 1989.
[3] S. Pincus and R. E. Kalman, “Not all (possibly) “random” sequences are
created equal” PNAS, 94(8): 3513-3518, April 15, 1997.
(A) No sequence can be too evenly
distributed?
• If s1, s2, … is an infinite sequence of real
numbers in (0,1), then corresponding to any
arbitrary large K, there exist a positive integer
n and two subintervals, of equal length, of the
interval (0,1), such that the number of sv with
v=1,2,…,n that lie in one of the subintervals
differs from the number of such sv that lie in
the other subinterval by more than K.
(B) van Aardenne-Ehrenfest’s refinement
• Let A = (s1, s2, s3, … , sN) be a sequence in U = [0,1).
For any integer 1≤n≤N & real number 0<α<1, let
D( A, n,  )  ( 1)  n  Dn ( )  n
0 si 
1i  n
(B) van Aardenne-Ehrenfest’s refinement
• Let A = (s1, s2, s3, … , sN) be a sequence in U = [0,1).
For any integer 1≤n≤N & real number 0<α<1, let
D( A, n,  )  ( 1)  n  Dn ( )  n
0 si 
1i  n
& discrepancy:
D( A, n)  sup | D( A, n,  ) |
0 1
(B) van Aardenne-Ehrenfest’s refinement
• Let A = (s1, s2, s3, … , sN) be a sequence in U = [0,1).
For any integer 1≤n≤N & real number 0<α<1, let
D( A, n,  )  ( 1)  n  Dn ( )  n
0 si 
1i  n
& discrepancy:
D( A, n)  sup | D( A, n,  ) |
0 1
Recall: the sequence A is called uniformly distributed
(on [0, 1)) if D(N) is o(N) [3].
(B) van Aardenne-Ehrenfest’s refinement
• Let A = (s1, s2, s3, … , sN) be a sequence in U = [0,1).
For any integer 1≤n≤N & real number 0<α<1, let
D( A, n,  )  ( 1)  n  Dn ( )  n
0 si 
1i  n
& discrepancy:
Then
D( A, n)  sup | D( A, n,  ) |
0 1
log log N
D( A, n) 
log log log N
for infinitely many n[2].
(B) van Aardenne-Ehrenfest’s refinement
• Let A = (s1, s2, s3, … , sN) be a sequence in U = [0,1).
For any integer 1≤n≤N & real number 0<α<1, let
D( A, n,  )  ( 1)  n  Dn ( )  n
0 si 
1i  n
& discrepancy:
Then
D( A, n)  sup | D( A, n,  ) |
0 1
log log N
D( A, n) 
log log log N
for infinitely many n[2].
Discrepancy D(A,n) >> ? [2]
• In 1949 Mrs. van Aardenne-Ehrenfest:
log log N
D( A, n) 
log log log N
Discrepancy D(A,n) >> ? [2]
• In 1949 Mrs. van Aardenne-Ehrenfest:
log log N
D( A, n) 
log log log N
• In 1954 Roth
D( A, n)  (log N )1/ 2
Discrepancy D(A,n) >> ? [2]
• In 1949 Mrs. van Aardenne-Ehrenfest:
log log N
D( A, n) 
log log log N
• In 1954 Roth
D( A, n)  (log N )1/ 2
• In 1972 Schmidt
D( A, n)  log N
Discrepancy D(A,n) >> ? [2]
• In 1949 Mrs. van Aardenne-Ehrenfest:
log log N
D( A, n) 
log log log N
• In 1954 Roth
D( A, n)  (log N )1/ 2
• In 1972 Schmidt
D( A, n)  log N
(C) Equivalent form of (B)
• Let N be a large integer , and let P1, P2, …, PN
be points, not necessarily distinct, in the
square 0≤x≤1, 0≤y≤1. For any point (u,v) in
this square, let S(u,v) denote the number of
points in the rectangle 0≤x<u, 0≤y<v. Then
there exist x0 & y0, with 0<x0<1, 0<y0<1, such
that
log log N
| S ( x0 , y0 )  Nx0 y0 |
log log log N
(C) Equivalent form of (B) (cont.)
• Roth’s idea behind:
– (B)
log log N
| Dn ( )  n |
log log log N
– (C) | S ( x , y )  Nx y | log log N
0
0
0 0
log log log N
(C) Equivalent form of (B) (cont.)
• Roth’s idea behind:
log log N
| Dn ( )  n |
log log log N
– (B)
– (C) | S ( x , y )  Nx y | log log N
0
0
0 0
log log log N
• Recall:
D( A, n,  )  (
1)  n  D ( )  n
0 si 
1i  n
n
Roth’s theorem
• Roth’s idea behind:
– (B)
log log N
| Dn ( )  n |
log log log N
– (C) | S ( x , y )  Nx y | log log N
0
0
0 0
log log log N
• Roth’s theorem:
1 1

0 0
(S ( x, y)  Nxy) 2 dxdy  log N
Notation
• Representation of 0≤x<1:
x
x1 ( x) x2 ( x) x3 ( x)
 2  3  ...,
2
2
2
xi ( x)  0 or 1
• r ( x)  (1) x , for r  1,2,..., n  1
r
• Sign function

 x ( X )  x1 ( x),..., xr 1 ( X v )  xr 1 ( x);
0,
if  v for which  1 v

Fr ( x, y )  
 y1 (Yv )  y1 ( y ),..., yn r 1 (Yv )  yn r 1 ( y );
 ( x) ( y ), if ~  such v
nr
 r
n 1
•
F ( x, y )   Fr ( x, y )
r 1
Lemmas
• Lemma 1.
1 1

0 0
xyF( x, y)dxdy  (n  1) 22 n (2n2  N )
• Lemma 2.
1 1

2
{
F
(
x
,
y
)}
dxdy  n  1

0 0
• Lemma 3.
1
1
xv
yv
 
F ( x, y )dxdy  0
• Schwarz’s inequality
2
2
|  f ( x, y ) g ( x, y )dxdy |   | f ( x, y ) | dxdy   | g ( x, y ) | dxdy
2
A “well-distributed” set of 2D points
• Construction
– let P1 ,..., P2 be the 2n points of the form:
n
tn tn
t1
t1
(  ... n ,  ... n )
2
2 2
2
• Property
– Each P lies in the square [0,0]≤(x,y)≤[1,1];
–
| S ( x, y)  2n xy | 2n  1
Thank you!