An Elementary View of Weyl's Theory of Equal Distribution

Suppose that −∞ < a < b < ∞, a ≤ u1n ≤ u2n ≤ · · · ≤ unn ≤ b, and a ≤ v1n ≤ v2n ≤ · · · ≤ vnn ≤ b for n ≥ 1. We simplify and strengthen Weyl’s definition of equal distribution of {{uin} n i=1} ∞ n=1 and {{vin} n i=1} ∞ n=1 by showing that the following statements are equivalent: (i) limn→∞ 1 n ∑n i=1 (F (uin) − F (vin)) = 0 for all F ∈ C[a, b], (ii) limn→∞ 1 n ∑n i=1 |uin − vin| = 0, (iii) limn→∞ 1 n ∑n i=1 |F (uin)− F (vin)| = 0 for all F ∈ C[a, b]. We relate this to Weyl’s definition of uniform distribution and Szegö’s distribution formula for the eigenvalues of a family of Toeplitz matrices {[tr−s] n r,s=1} ∞ n=1, where tr = 1 2π ∫ π −π eg(x) dx and g is real-valued and continuous on [−π, π].