Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$ and $\alpha > 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus ${\mathbb T}^d$ or the $d$-dimensional unit cube ${\mathbb I}^d$. We prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb T}^2)$ in the case $1\leq p \leq \infty$, $0 < \theta \leq \infty$, $\alpha>1/p$. A non-periodic modification of this classical formula yields upper bounds for $B^{\alpha}_{p,\theta}({\mathbb I}^2)$ if $1/p<\alpha<1+1/p$. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that a corresponding result is most likely also true in case $d>2$. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.