Israel Journal of Mathematics

, Volume 157, Issue 1, pp 219–238 | Cite as

Pair correlations of sequences in higher dimensions

  • R. Nair
  • M. Pollicott
Article

Abstract

We consider a system of “generalised linear forms” defined on a subset x = (x ij ) of ℝd by
$$ L_1 (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k) = \sum\limits_{j = 1}^{d_1 } {g_{1j}^k (x_{1j} ), \ldots ,} L_l (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k) = \sum\limits_{j = 1}^{d_l } {g_{lj}^k (x_{lj} ) \in \mathbb{R}} , for k \geqslant 1, $$
, where d = d 1 + ⋯+ d l and for each pair of integers (i, j), 1 ≤ il, 1 ≤ jd i the sequence of functions (g ij k (x)) k=1 is differentiable on an interval X ij . Then let
$$ X_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) = (\{ L_1 (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k)\} , \ldots ,\{ L_l (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k)\} ) \in \mathbb{T}^l , $$
, for x in the Cartesian product X = × i=1 l × j=1 di X ij ⊂ ℝd. Let R = I 1 × ⋯ × I l be a rectangle in \( \mathbb{T}^l \) and for each N ≥ 1 let
$$ V_N (R) = \sum\limits_{1 \leqslant n \ne m \leqslant N} {\chi _R (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} _n (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} _m (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ))} $$
and then define
$$ \Delta _N = \mathop {\sup }\limits_{R \subset \mathbb{T}^l } \{ V_N (R) - N(N - 1) leb(R)\} $$
where the supremum is over all rectangles in \( \mathbb{T}^l \). We show that for almost every x\( \mathbb{T}^d \) we have that
$$ \Delta _N = O(N(\log N)^\alpha ), $$
for appropriate α. Other related results are also described.

Keywords

High Dimension Pair Correlation London Mathematical Society Dependent Random Variable Half Open Interval 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BTP]
    I. Berkes, W. Philipp and R. Tichy, Pair correlations and U-statistics for independent and weakly dependent random variables, Illinois Journal of Mathematics 45 (2001), 559–580.MATHMathSciNetGoogle Scholar
  2. [BP]
    I. Berkes and W. Philipp, The size of trigonometric and Walsh series and uniform distribution mod 1, Journal of the London Mathematical Society (2) 50 (1994), 454–464.MATHMathSciNetGoogle Scholar
  3. [C]
    J. W. S. Cassels, Some metrical theorems of Diophantine approximation. II, Journal of the London Mathematical Society 25 (1950), 180–184.MATHCrossRefMathSciNetGoogle Scholar
  4. [H]
    R. A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill 1967), Southern Illinois University Press, Carbondale, Ill, 1968, pp. 235–255.Google Scholar
  5. [KN]
    L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, Wiley-Interscience, New York, 1974.Google Scholar
  6. [L]
    W. J. LeVeque, The distribution modulo 1 of trigonometric sequences, Duke Mathematical Journal 20 (1953), 367–374.MATHCrossRefMathSciNetGoogle Scholar
  7. [MV]
    H. L. Montgomery and J. D. Vaaler, Maximal variants of basic inequalities, in Proceedings of the Congress on Number Theory (Spanish) (Zarauz, 1984), University Pas Vasco-Euskal Herriko Unib., Bilbao, 1989, pp. 181–197.Google Scholar
  8. [Na1]
    R. Nair, Some theorems on metric uniform distribution using L 2 methods, Journal of Number Theory 35 (1990), 18–52.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • R. Nair
    • 1
  • M. Pollicott
    • 2
  1. 1.Department of MathematicsUniversity of LiverpoolLiverpoolUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations