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