Journal of Combinatorics
Volume 1 (2010)
Lattice point counting and the probabilistic method
Pages: 171 – 232
We take a quantitative approach based on probability theory to several number theoretic problems, which all have the common form of counting lattice points in some nice domain. It is well-known that the number of solutions to Pell equations can be counted with a bounded error term. We relax Pell equations to (inhomogeneous) Pell inequalities and study the corresponding question. A naive area principle (=the number of lattice points in and the area of nice domains are close) guides the intuition for the answer, but the intuition is sometimes correct, sometimes not. On the one hand, the intuition fails for continuum many translated copies of the corresponding hyperbolic domain (Theorem 1). On the other hand, the intuition is correct for almost all translated copies (Theorems 2 and 3).