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, 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.
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Beck, J. Pell equation and randomness. Period Math Hung 70, 1–108 (2015). https://doi.org/10.1007/s10998-014-0064-x
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DOI: https://doi.org/10.1007/s10998-014-0064-x
Keywords
- Lattice point counting in specified regions
- Discrepancy
- Central limit theorem
- Law of the iterated logarithm