On the correlations of directions in the euclidean plane

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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 languageEnglish (US)
Pages (from-to)1797-1825
Number of pages29
JournalTransactions of the American Mathematical Society
Issue number4
StatePublished - Apr 2006


  • Correlation measures
  • Directions in ℝ

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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