Discrete universality of the Riemann zeta-function and uniform distribution modulo 1

It is proved that a wide class of analytic functions can be approximated by shifts $\zeta(s+i\varphi(k))$, $k\geqslant k_0$, $k\in\mathbb N$, of the Riemann zeta-function. Here the function $\varphi(t)$ has a continuous nonvanishing derivative on $[k_0,\infty)$ satisfying the estimate $\varphi(2t)\max_{t\leqslant u\leqslant2t}(\varphi'(u))^{-1}\ll t$, and the sequence $\{a\varphi(k)\colon k\geqslant k_0\}$ with every real $a\neq0$ is uniformly distributed modulo 1. Examples of $\varphi(t)$ are given.