Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square Authors: Dinh Dũng, Tino Ullrich We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d-dimensional torus or the d-dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci-lattice for bivariate periodic functions from in the case , and . A non-periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formul...