On differentiation of integrals in the infinite-dimensional torus

We answer the recently posed questions regarding the problem of differentiation of integrals for the Rubio de Francia basis $\mathcal{R}$ in the infinite torus ${\mathbb{T}}^{\omega }$. In particular, we prove that $\mathcal{R}$ does not differentiate $L^{\mathrm{\infty }}({\mathbb{T}}^{\omega })$. Some remarks about differentiation in the context of arbitrary bases are also included.