On uniformity of $q$-multiplicative sequences

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$($\alpha \in $\mathbb{R}$)$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\pi i p(n)}$($p \in $\mathbb{R}$[x]$)\mathrm{.}Quantitatively,weshowthatany$q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.