On Strong Uniform Distribution Iv

Let a= (ai)i=1 be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We say that a is an ∗ sequence if for each f ∈ we set AN ( f ,x) = (1/N) ∑N i=1 f ({aix}) (N = 1,2, . . .), then limN→∞AN ( f ,x) = ∫ 1 0 f (t)dt, almost everywhere with respect to Lebesgue measure. LetVq( f ,x) = ( ∑∞ N=1 |AN+1( f ,x)−AN ( f ,x)|q)1/q (q ≥ 1). In this paper, we show that if a is an (Lp)∗ for p > 1, then there exists Dq > 0 such that if ‖ f ‖p denotes ( ∫ 1 0 | f (x)|pdx)1/p, ‖Vq( f ,·)‖q ≤ Dq‖ f ‖p (q > 1). We also show that for any (L1)∗ sequence a and any nonconstant integrable function f on the interval [0,1), V1( f ,x) =∞, almost everywhere with respect to Lebesgue measure.