On the theorem of Davenport and generalized Dedekind sums

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.