### Abstract

Let f(φ) be a positive continuous function on 0 ≤ φ ≤ Θ, where Θ ≤ 2π, and let ξ be the number of two-dimensional lattice points in the domain Π_{R}(f) between the curves r = (R + c_{1}/R)f(φ) and r = (R + c_{2}/R)f(φ), where c_{1} < c_{2} are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ_{L} on the interval [a_{1}L, a_{2}L], Sinai showed that the distribution of ξ under P × μ_{L} converges to a mixture of the Poisson distributions as L → ∞. Later Major showed that for P-almost all f, the distribution of ξ under μ_{L} converges to a Poisson distribution as L → ∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai.

Original language | English |
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Pages (from-to) | 203-247 |

Number of pages | 45 |

Journal | Communications in Mathematical Physics |

Volume | 213 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2000 Sep |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics