On Irregularities of Distribution

1. 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.

2. (A) No sequence can be too evenly distributed? • If s1, s2, … is an infinite sequence of real numbers in (0,1), thencorresponding 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.

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

4. (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 & discrepancy:

5. (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 & discrepancy: Recall: the sequence A is called uniformly distributed (on [0, 1)) if D(N) is o(N)[3].

6. (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 & discrepancy: Then for infinitely many n[2].

7. (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 & discrepancy: Then for infinitely many n[2].

8. Discrepancy D(A,n) >> ? [2] • In 1949 Mrs. van Aardenne-Ehrenfest:

9. Discrepancy D(A,n) >> ? [2] • In 1949 Mrs. van Aardenne-Ehrenfest: • In 1954 Roth

10. Discrepancy D(A,n) >> ? [2] • In 1949 Mrs. van Aardenne-Ehrenfest: • In 1954 Roth • In 1972 Schmidt

11. Discrepancy D(A,n) >> ? [2] • In 1949 Mrs. van Aardenne-Ehrenfest: • In 1954 Roth • In 1972 Schmidt

12. (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

13. (C) Equivalent form of (B) (cont.) • Roth’s idea behind: • (B) • (C)

14. (C) Equivalent form of (B) (cont.) • Roth’s idea behind: • (B) • (C) • Recall:

15. Roth’s theorem • Roth’s idea behind: • (B) • (C) • Roth’s theorem:

16. Notation • Representation of 0≤x<1: • Sign function

17. Lemmas • Lemma 1. • Lemma 2. • Lemma 3. • Schwarz’s inequality

18. A “well-distributed” set of 2D points • Construction • let be the 2n points of the form: • Property • Each P lies in the square [0,0]≤(x,y)≤[1,1];

19. Thank you!