On strong uniform distribution III

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...

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Veröffentlicht in:Indagationes mathematicae Jg. 14; H. 2; S. 233 - 240
1. Verfasser: Nair, R.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 23.06.2003
ISSN:0019-3577, 1872-6100
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Abstract 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.
AbstractList 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.
Author Nair, R.
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Cites_doi 10.1007/PL00010091
10.1007/BF02566244
10.2307/1970516
10.4064/aa-56-3-183-193
10.1007/BF01470064
10.1007/BF01475864
10.4064/sm-42-3-271-288
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