On strong uniform distribution III - Nair, R. (Indagationes mathematicae, 2003)
Let a = ( a i i=1 ∞ be a strictly increasing sequence of natural numbers and let A 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 A ∗ sequence if for each f ϵ A lim N→∞ 1 N ∑ i=1 N f(<a ix>)= ∫ 0 1 f(t)dt, almost everywhere with respect to Lebesgue measure. Let a i=(a 1,i) ∞ i=1, …,a k=(a k,i) ∞ i=1, denote finitely many ( L 1) ∗ sequences, and for a sequence a, let G a(u)=|{i:a i ≤ u}|, where for a finite set A we have used | A| to denote its cardinality. Also let a 1 ∘ … ∘ a k denote the set {b 1…b k : b 1ϵa k}, counted with multiplicity and ordered by absolute value. Suppose there exists K > 0 such that for all u ≥ 1 |G a 1 (u)|…|G a k (u)|≤|G a 1 compfn;…compfn; a k (u)|. Then if log +| x| = log max(1,| x|) we show that a 1 ∘ … ∘ a k is an ( L( log + L) k−1 ) ∗ sequence.