In a recent joint paper with J. Schoißengeier we described an equivalent condition for the approximation of an
improper (Riemann-) integral using uniformly distributed sequences modulo 1. (We used the sequence of multiples
of an irrational number modulo 1.) Our results were subject to the restriction that the singularities of the integrand
had to heave rational coordinates. The present project`s aim is to remove and/or explain this restriction. There are
crucial connections to Diophantine approximation as it is of great importance how well the singularities can be
approximated by the terms of the sequence.