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We prove that, for every rational $d\ne 0,\pm 1$ and every compact set$K\subset\{s\in\mathbb{C}:1/2<\Re(s)<1\}$ with connected complement, anyanalytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated,uniformly on $K$, by the shifts $\zeta(s+i\tau)$ and $\zeta(s+id\tau)$,respectively. As a consequence we deduce that the set of $\tau$ satisfying$|\zeta(s+i\tau)-\zeta(s+id\tau)|<\varepsilon$ uniformly on $K$ has a positivelower density for every $d\ne 0$.

Joint universality and generalized strong recurrence with rational parameter