Discrepancy Estimate of Normal Vectors

is called the discrepancy of (xn)n=1. In 1954 Roth (see [DrTi], [KN]) proved that for any sequence in [0, 1) limN→∞ND(N)/ log N > 0. (2) Let A be an s × s invertible matrix with integer entries. A matrix A is said to be ergodic if for almost all α ∈ R the sequence {αA}n≥1 is uniformly distributed. A vector α ∈ R is said to be normal (A normal) if the sequence {αA}n≥1 is uniformly distributed. Let λi (1 ≤ i ≤ s) denote the eigenvalues of a matrix A. For the case of |λi| > 1, i = 1, . . . , s normal vectors were constructed by Postnikov (s = 2) and by Polosuev (s ≥ 2) (see [Po]). Normal vectors were constructed for the general