Uniform distribution of sequences

( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less than x. The ollowing distribution properties of the sequence (Xn)=(nO)(t an arbitrary positive real number) are well-known" (i) I Z--Zn_c and Z/Zn_xol as n-c, then (Xn) is uniformly distributed mod z (W. J. Le Veque [6]). (ii) If z-z_x is decreasing, then (Xn)is uniformly distributed modz for almost all t; this result also holds in the case (Xn)=(nr) for any fixed ’0 (H. Davenport and W. J. Le Veque [3]). (iii) If z/z_--.1 as n--c and if the number of terms z with znN is less than c.N(c, a>0), then (x) is uniformly distributed modz for almost all t (H. Davenport and P. ErdSs [2]). In the following we prove a generalization of some of these results by an elementary method (cf. [7]). For this purpose we define a sequence (x) to be almost uniformly distributed mod z if there is an infinite sequence N<N<... of positive integers such that ( 3 ) lim (1/N)A(x, N, {x})= x (0x-<l)