A hybrid inequality of Erd\"os-Tur\'an-Koksma for digital sequences

For bases $\mathbf{b}=(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$, we prove a version of the inequality of \etk \ for the hybrid function system composed of the Walsh functions in base ${\text{bfb}}^{(1)}=(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_{s_1})$ and, as second component, the ${\text{bfb}}^{(2)}$-adic functions, ${\text{bfb}}^{(2)}=(b_{s_1+1},\mathrm{.}\mathrm{.}\mathrm{.},b_s)$, with $s=s_1+s_2$, $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.