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### Pair correlations of sequences in higher dimensions

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Nair, R and Pollicott, Mark.
(2007)
*Pair correlations of sequences in higher dimensions.*
Israel Journal of Mathematics, Volume 157
(Number 1).
pp. 219-238.
ISSN 0021-2172

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1007/s11856-006-0009-z

## Abstract

We consider a system of "generalised linear forms" defined on a subset x = (x(ij)) of R-d by

L-1 (x) (k) = d(1) g(1j)(k) (x(1j)), ...L-l (x) (k) = d(l) g(lj)(k) (xlj) is an element of R, for k >= 1,

where d = d(1) + ...+ d(l) and for each pair of integers (i, j), 1 <= i <= l l <= j <= d(i) the sequence of functions (g(ij)(k)(x))(kappa=1)(infinity) is differentiable on an interval X-ij. Then let

X-K(x) = ({L-1(x)(k)}, . . . , {L-l(x)(k)}) is an element of T-l

for x in the Cartesin product X = x(i=1)(l) x (di)(j = 1) X-ij subset of R-d. Let R = I-1 x . . . x I-l be a rectangle in T-l and for each N >= 1 let

VN(R) = Sigma(1 <= n not equal m <= N) XR(X-n)(X)-(X-m)(x))

and then define

Delta(N) = sup {V-N (R) - N (N-1) leb (R) } R subset of T-l

where the supremum is over all rectangles in T-l. We show that for almost every x is an element of T-d we have that

Delta(N) = O(N(log N)(alpha))

for appropiate alpha. Other related results are also described.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Journal or Publication Title: | Israel Journal of Mathematics |

Publisher: | Magnes Press |

ISSN: | 0021-2172 |

Official Date: | January 2007 |

Volume: | Volume 157 |

Number: | Number 1 |

Number of Pages: | 20 |

Page Range: | pp. 219-238 |

Identification Number: | 10.1007/s11856-006-0009-z |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

URI: | http://wrap.warwick.ac.uk/id/eprint/31717 |

Data sourced from Thomson Reuters' Web of Knowledge

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