Poisson law for the number of lattice points in a random strip with finite area

Summary Let a smooth curve be given by a functionr=f() in polar coordinate system in the plane, and letR be a uniformly distributed random variable on the interval [a₁L, a₂L] with somea₂>a₁>0 and a largeL>0. Ya. G. Sinai has conjectured that given some real numbersc₂>c₁, the number of lattice points in the domain between the curves and is asymptotically Poisson distributed for good functionsf(·). We cannot prove this conjecture, but we show that if a probability measure with some nice properties is given on the space of smooth functions, then almost all functions with respect to this measure satisfy Sinai's conjecture. This is an improvement of an earlier result of Sinai [9], and actually the proof also contains many ideas of that paper.This article was processed by the author using the Springer-Verlag T_EX ProbTh macro package 1991.