On the theorem of Davenport and generalized Dedekind sums

TL;DR

This paper analyzes the discrepancy of symmetrized lattice point sets in the unit square related to irrational numbers, especially quadratic irrationals like the golden ratio, and connects these results to generalized Dedekind sums.

Contribution

It provides explicit formulas for the $L^2$ discrepancy of lattice points based on the properties of the irrational number and introduces methods to approximate generalized Dedekind sums using partial quotients.

Findings

For quadratic irrationals, the discrepancy grows as $c(\alpha) \log n$.

The golden ratio minimizes the $L^2$ discrepancy among explicit constructions.

Generalized Dedekind sums can be approximated using partial quotients of rational arguments.

Abstract

A symmetrized lattice of $2n$ points in terms of an irrational real number $\alpha$ is considered in the unit square, as in the theorem of Davenport. If $\alpha$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be $c( \alpha ) \log n + O \left( \log \log n \right)$ for a computable positive constant $c( \alpha )$. For the golden ratio $\varphi$, the value $\sqrt{c (\varphi ) \log n}$ yields the smallest $L^2$ discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients $a_k$ of $\alpha$ grow at most polynomially fast, the $L^2$ discrepancy is found in terms of $a_k$ up to an explicitly bounded error term. It is also shown that certain generalized Dedekind sums can be approximated using the same methods. For a special generalized Dedekind sum with arguments $a, b$ an asymptotic formula in terms of the partial quotients of $\frac{a}{b}$ is proved.