Metric number theory, lacunary series and systems of dilated functions Christoph Aistleitner aistleitner@math.tugraz.at Kobe University, Japan Uniform distribution and quasi-Monte Carlo methods RICAM Linz, Austria October 14, 2013
The equidistribution theorem The sequence of fractional parts ( { kx } ) k ≥ 1 is uniformly distributed modulo 1 if and only if x �∈ Q (Bohl–Sierpinski–Weyl, 1909–1910). A sequence ( x k ) k ≥ 1 is called uniformly distributed modulo 1 if for all a ∈ [0 , 1] the relation N 1 � lim 1 [0 , a ) ( x k ) = a N N →∞ k =1 holds. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
The equidistribution theorem The sequence of fractional parts ( { kx } ) k ≥ 1 is uniformly distributed modulo 1 if and only if x �∈ Q (Bohl–Sierpinski–Weyl, 1909–1910). A sequence ( x k ) k ≥ 1 is called uniformly distributed modulo 1 if for all a ∈ [0 , 1] the relation N 1 � lim 1 [0 , a ) ( x k ) = a N N →∞ k =1 holds. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
The equidistribution theorem For a sequence ( n k ) k ≥ 1 of distinct integers, the sequence of fractional parts ( { n k x } ) k ≥ 1 is uniformly distributed modulo 1 for almost all x . (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for n k = 2 k , k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy . Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
The equidistribution theorem For a sequence ( n k ) k ≥ 1 of distinct integers, the sequence of fractional parts ( { n k x } ) k ≥ 1 is uniformly distributed modulo 1 for almost all x . (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for n k = 2 k , k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy . Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
The equidistribution theorem For a sequence ( n k ) k ≥ 1 of distinct integers, the sequence of fractional parts ( { n k x } ) k ≥ 1 is uniformly distributed modulo 1 for almost all x . (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for n k = 2 k , k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy . Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
The equidistribution theorem For a sequence ( n k ) k ≥ 1 of distinct integers, the sequence of fractional parts ( { n k x } ) k ≥ 1 is uniformly distributed modulo 1 for almost all x . (Weyl, 1916) In general the exceptional set cannot be determined explicitly (for example for n k = 2 k , k ≥ 1.) What is the speed of convergence in Weyl’s theorem? – The answer is given in terms of the discrepancy . Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Discrepancy For a finite sequence x 1 , . . . , x N of points in the unit interval the star-discrepancy D ∗ N is defined as � � N 1 � � � D ∗ N ( x 1 , . . . , x N ) = sup 1 [0 , a ) ( x k ) − a � . � � N � � 0 ≤ a ≤ 1 � k =1 We want to find upper bounds for D ∗ N ( { n 1 x } , . . . , { n N x } ) N → ∞ . as Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Discrepancy For a finite sequence x 1 , . . . , x N of points in the unit interval the star-discrepancy D ∗ N is defined as � � N 1 � � � D ∗ N ( x 1 , . . . , x N ) = sup 1 [0 , a ) ( x k ) − a � . � � N � � 0 ≤ a ≤ 1 � k =1 We want to find upper bounds for D ∗ N ( { n 1 x } , . . . , { n N x } ) N → ∞ . as Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Closely related problems Find upper bounds for � N � � � � 1 [0 , a ) ( { n k x } ) − Na � � � � � � k =1 for fixed a and a.e. x . Find upper bounds for � N � � � � f ( n k x ) � � � � � � k =1 for f being centered, 1-periodic and of bounded variation, for a.e. x . Find criteria for the a.e. convergence of ∞ � c k f ( n k x ) k =1 for f being centered, 1-periodic and of bounded variation. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Closely related problems Find upper bounds for � N � � � � 1 [0 , a ) ( { n k x } ) − Na � � � � � � k =1 for fixed a and a.e. x . Find upper bounds for � N � � � � f ( n k x ) � � � � � � k =1 for f being centered, 1-periodic and of bounded variation, for a.e. x . Find criteria for the a.e. convergence of ∞ � c k f ( n k x ) k =1 for f being centered, 1-periodic and of bounded variation. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Closely related problems Find upper bounds for � N � � � � 1 [0 , a ) ( { n k x } ) − Na � � � � � � k =1 for fixed a and a.e. x . Find upper bounds for � N � � � � f ( n k x ) � � � � � � k =1 for f being centered, 1-periodic and of bounded variation, for a.e. x . Find criteria for the a.e. convergence of ∞ � c k f ( n k x ) k =1 for f being centered, 1-periodic and of bounded variation. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks I: Known results In a few special cases the answers are known. If n k = k , k ≥ 1 (connection with continued fractions expansion). If ( n k ) k ≥ 1 is increasing very quickly ( lacunary sequences - almost independent behavior). Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks I: Known results In a few special cases the answers are known. If n k = k , k ≥ 1 (connection with continued fractions expansion). If ( n k ) k ≥ 1 is increasing very quickly ( lacunary sequences - almost independent behavior). Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks I: Known results In a few special cases the answers are known. If n k = k , k ≥ 1 (connection with continued fractions expansion). If ( n k ) k ≥ 1 is increasing very quickly ( lacunary sequences - almost independent behavior). Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks II: Probabilistic interpretation It suffices to consider x ∈ [0 , 1]. The space ([0 , 1] , B , λ ) is a probability space. The functions { n k x } and f ( n k x ) are random variables . For each k , the random variable { n k x } has distribution U ([0 , 1]). The discrepancy is the Kolmogorov–Smirnov statistic. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks II: Probabilistic interpretation It suffices to consider x ∈ [0 , 1]. The space ([0 , 1] , B , λ ) is a probability space. The functions { n k x } and f ( n k x ) are random variables . For each k , the random variable { n k x } has distribution U ([0 , 1]). The discrepancy is the Kolmogorov–Smirnov statistic. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks II: Probabilistic interpretation It suffices to consider x ∈ [0 , 1]. The space ([0 , 1] , B , λ ) is a probability space. The functions { n k x } and f ( n k x ) are random variables . For each k , the random variable { n k x } has distribution U ([0 , 1]). The discrepancy is the Kolmogorov–Smirnov statistic. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks II: Probabilistic interpretation It suffices to consider x ∈ [0 , 1]. The space ([0 , 1] , B , λ ) is a probability space. The functions { n k x } and f ( n k x ) are random variables . For each k , the random variable { n k x } has distribution U ([0 , 1]). The discrepancy is the Kolmogorov–Smirnov statistic. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Remarks II: Probabilistic interpretation It suffices to consider x ∈ [0 , 1]. The space ([0 , 1] , B , λ ) is a probability space. The functions { n k x } and f ( n k x ) are random variables . For each k , the random variable { n k x } has distribution U ([0 , 1]). The discrepancy is the Kolmogorov–Smirnov statistic. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Tools I Erd˝ os–Tur´ an inequality: For any positive integer H � � H N N ( x 1 , . . . , x N ) ≤ 3 1 1 � � � � e 2 π ihx k D ∗ H + 3 � . � � h N � � � h =1 k =1 Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
Tools II Koksma’s inequality: For any function f which has bounded variation Var( f ) in the unit interval we have � 1 � � N 1 � � � � ≤ Var( f ) · D ∗ f ( x k ) − f ( x ) dx N ( x 1 , . . . , x N ) . � � N � � 0 � k =1 Christoph Aistleitner Metric number theory, lacunary series and systems of dilated func
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