ON THE NUMBER OF LATTICE POINTS BETWEEN 2 ENLARGED AND RANDOMLY SHIFTED COPIES OF AN OVAL

• 成果类型: Article
• 署名作者: CHENG, ZM; LEBOWITZ, JL; MAJOR, P
• 署名单位: Rutgers University System; Rutgers University New Brunswick; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Hungarian Academy of Sciences
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01199268
• 发表日期: 1994
• 页码: 253-268
• 关键词: quantum chaos

摘要

Let A be an oval with a nice boundary in R2, R a large positive number, c > 0 some fixed number and alpha a uniformly distributed random vector in the unit square [0, 1]2. We are interested in the number of lattice points in the shifted annular region consisting of the difference of the sets {(R + c/R) A - alpha} and {(R - c/R) A - alpha}. We prove that when R tends to infinity, the expectation and the variance of this random variable tend to 4c times the area of the set A, i.e. to the area of the domain where we are counting the number of lattice points. This is consistent with computer studies in the case of a circle or an ellipse which indicate that the distribution of this random variable tends to the Poisson law. We also make some comments about possible generalizations.