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

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 vec tor in the unit square [0, 1]2. We are interested in the number of lat tice 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 th is 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 variabl e tends to the Poisson law. We also make some comments about possible generalizations.