We simulate lattice models in the three-dimensional XY universality class inthe low and the high temperature phase. This allows us to compute a number ofuniversal amplitude ratios with unprecedented precision: R_{\Upsilon}=0.411(2),R_B=2.83(1), R_{\xi}^+=0.3562(10) and R_{\xi}^-=0.850(5). These results can becompared with those obtained from other theoretical methods, such as fieldtheoretic methods or the high temperature series expansion and also withexperimental results for the lambda-transition of $^4$He. In addition to the XYmodel, we study the three-dimensional two-component $\phi^4$ model on thesimple cubic lattice. The parameter of the $\phi^4$ model is chosen such thatleading corrections to scaling are small.
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