We prove two uniform weighted ergodic theorems for \(C^*\) -dynamical systems. If \(({\mathfrak {A}},\alpha )\) is uniquely ergodic relative to \({\mathfrak {A}}^\alpha \) then for any class M of phases with order m difference control at rate \(\delta \in (0,1)\) the averages \( \frac{1}{N}\sum _{n\le N} e\!\big (p(n)\big )\,\alpha ^n(a)\rightarrow 0 \ \ \text {uniformly in} \ p\in M. \) for every \(a\in \ker E\) with norm rate \(O(N^{-\delta })\) . Moreover, if \(({\mathfrak {A}},\alpha )\) is uniformly weakly mixing in norm along positive-density subsequences and M also enjoys uniform cancellation, then the same uniform convergence holds for all \(a\in {\mathfrak {A}}\) along any positive-density subsequence. We then analyze Anzai skew- products on the rotation algebra and give a concrete description of relative unique ergodicity to which the obtained results can be applied.
