## Abstract

Let R _{(x,y),Q} ^{(v)} denote the repartition of the v-level correlation measure of the finite set of directions P _{(x,y)}P, where P _{(x,Y)} is the fixed point (x, y) ∈ [0, 1) ^{2} and P is an integer lattice point in the square [-Q, Q] ^{2}. We show that the average of the pair correlation repartition R _{(x,y),Q} ^{(2)} over (x, y) in a fixed disc double-struck D sign _{0} converges as Q → ∞. More precisely we prove, for every λ ∈ ℝ _{+} and 0 < δ < 1/10, the estimate 1/Area(double-struck D sign _{0}) ∫∫D _{0} R _{(x,y),Q} ^{(2)}(λ)dx dy = 2πλ/3 + Odouble-struck D sign _{0,λ,δ}(Q ^{-1/10+δ}) as Q → ∞. We also proves that for each individual point (x, y) ∈ [0, 1) ^{2}, the 6-level correlation R _{(x,y),Q} ^{(6)} (λ) diverges at any point λ ∈ ℝ _{+} ^{5} as Q → ∞, and we give an explicit lower bound for the rate of divergence.

Original language | English (US) |
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Pages (from-to) | 1797-1825 |

Number of pages | 29 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2006 |

## Keywords

- Correlation measures
- Directions in ℝ

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics