Einige weitere Resultate in Analogie zu einem Gleichverteilungssatz von Koksma

Abstract

Continuing former investigations by the authors (see the references) the present paper contains metric results on the distribution modulo 1 of the powers of special kinds of real matricesA, namely of (2×2)- and (3×3)-triangle matrices, symmetric (2×2)-matrices and so-called “cosymmetric” (2×2)-matrices (i. e. matrices, symmetric with respect to the secondary diagonal). For almost all such matricesA (in the sense of the Lebesgue measure in ℝ³ resp. ℝ⁶) possessing no eigenvalue of modulus smaller than 1 the inequality

$$D(N) \leqslant C(A, \varepsilon ) N^{ - 1/2} (\log N)^{d + 3/2 + \varepsilon } (\varepsilon > 0)$$

is proved as an estimate for the discrepancy of the sequence (A ^s(n)) where (s(n)) ^=1/∞ _n is an arbitrary fixed strictly increasing sequence of positive integers andd is the dimension of the appropriate space ℝ^d (d=3 or 6).