Lattice Point Problems: Crossroads of Number Theory, Probability Theory and Fourier Analysis

Abstract

Diophantine approximation is a natural source of “lattice point counting” prob-lems. We count the number of lattice points in some “nice” shapes like tilted hyperbola seg-ments, tilted rectangles and axis-parallel right-angled triangles. The discrepancy from the “area” (i.e., expected value) depends heavily on the number-theoretic properties of the slope- in fact, it mainly depends on the continued fraction “digits” (called partial quotients) of the slope. Quadratic irrationals have the simplest (periodic) continued fractions, and this leads to quadratic fields, involving deep number theory. In the first two sections of this survey paper we study these kinds of topics. A key tool is Fourier analysis, and the big surprise is the un-expected appearance of probability theory which provides both deep insights and necessary tools. In the third section we switch from special shapes to arbitrary convex regions. In the fourth section we extend our investigations from the periodic set of lattice points to more gen-eral point distributions. Finally, the Appendix contains the proof of a lemma (which plays a key role in the third section).KeywordsCentral Limit TheoremDiophantine ApproximationQuadratic FieldPartial QuotientAsymptotic NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.