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 L2L2 discrepancy is found to be c(α)log⁡n+O(log⁡log⁡n)c(α)log⁡n+O(log⁡log⁡n) for a computable positive constant c(α)c(α). For the golden ratio φ  , the value c(φ)log⁡n yields the smallest L2L2 discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients akak of α   grow at most polynomially fast, the L2L2 discrepancy is found in terms of akak 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 ab is proved.