  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.
