Abstract
Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, where the slope is a quadratic irrational. Motivated by this relationship, we carry out here a systematic study of point counting with respect to translated or congruent families of any given long and narrow hyperbolic region. First we discuss the important special case when the underlying point set is the set of integer lattice points in the plane and the slope of the given hyperbolic region is arbitrary but fixed; see Theorems 3–21. Then we switch to the general case of an arbitrary point set of density one in the plane, and study point counting with respect to congruent copies of a given hyperbolic region; see Theorem 30. The main results are about the extra large discrepancy that we call superirregularity. This means that there is always a translated/congruent copy of any given long and narrow hyperbolic region of large area, for which the actual number of points in the copy differs from the area as much as possible, i.e. the discrepancy is at least a constant multiple of the area. Our theorems demonstrate, in a quantitative sense, that in point counting with respect to translated/congruent copies of any long and narrow hyperbolic region, superirregularity is inevitable.