Lattice point counting and the probabilistic method (Q543992)

The paper ``takes a quantitive 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.'' The author ``relaxes the Pell equations to (inhomogeneous) Pell inequalities and studies 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).'' Let \(F(\sqrt{2};\beta;\gamma;N)\) denote the number of integral solutions \((x,y)\in \mathbb{Z}^2\) of \[ -\gamma \leq (x+\beta)^2-2y^2\leq \gamma, \;\; \gamma>0, \beta\in [0,1) \] satisfying \(1\leq y\leq N\) and \(x\geq 1\). The theorems referred to above are: \textbf{Theorem 1.} For \(\gamma=1/2\) there are continuum many ``divergence points'' \(\beta^*\in [0,1)\) in the sense that \[ \limsup_{n\to\infty} \frac{F\left(\sqrt{2};\beta^*;\gamma=1/2;n\right)}{\log n}> \liminf_{n\to\infty} \frac{F\left(\sqrt{2};\beta^*;\gamma=1/2;n\right)}{\log n}. \] \textbf{Theorem 2.} Let \(\psi(x)\) be any positive decreasing function of the real variable \(x\) with \(\sum_n\psi(x)=\infty\). Then the inhomogeneous inequality \( \| n\alpha-\beta \|<\psi(n)\) has infinitely many integral solutions for almost all \(0\leq \beta <1\) (in the sense of Lebesgue measure). \textbf{Theorem 3.} Let \(\psi(x)\) be any positive decreasing function of the real variable \(x\) with \(\sum_n\psi(x)=\infty\). For any real number \(\alpha\), at least one of the following two cases always holds: {\parindent=8mm \begin{itemize}\item[(i)]the homogeneous inequality \(\| q\alpha \| < \psi(q)\) has infinitely many integral solutions, \item[(ii)]the inhomogeneous inequality \( \| q\alpha -\beta \| < \psi(q)\) has infinitely many integral solutions for almost all \(0\leq \beta <1\) (in the sense of Lebesgue measure). \end{itemize}}