Pell equation and randomness

作者：Beck Jzsef^*

来源：Periodica Mathematica Hungarica, 2015, 70(1): 1-108.

DOI：10.1007/s10998-014-0064-x

• Lattice point counting in specified regions
• Discrepancy
• Central limit theorem
• Law of the iterated logarithm

Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

• 出版日期2015-3
• 单位rutgers