Summary
Let A be an oval with a nice boundary in ℝ2,R a large positive number,c>0 some fixed number and α 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−α} and {(R−c/R)A−α}. We prove that whenR 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.
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Cheng, Z., Lebowitz, J.L. & Major, P. On the number of lattice points between two enlarged and randomly shifted, copies of an oval. Probab. Th. Rel. Fields 100, 253–268 (1994). https://doi.org/10.1007/BF01199268
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DOI: https://doi.org/10.1007/BF01199268