On the theorem of Davenport and generalized Dedekind sums

A symmetrized lattice of $2n$ points in terms of an irrational real number $α$ is considered in the unit square, as in the theorem of Davenport. If $α$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be $c( α) \log n + O \left( \log \log n \right)$ for a computable positive constant $c( α)$. For the golden ratio $φ$, the value $\sqrt{c (φ) \log n}$ 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 $α$ 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.

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