Zeros-distribution of the Riemann zeta-function and universality

In 1975, S.M. Voronin discovered the universality property of the zeta-function $\zeta(s)$, $s=\sigma+it$, i.e., that a wide class of analytic functions can be approximated by shifts $\zeta(s+i\tau)$, $\tau\in \mathbb{R}$.
We consider the universality of $\zeta(s)$ when $\tau$ takes values from the set $\{\gamma_{k}: k\in \mathbb{N}\}$, where $0<\gamma_{1}\le\gamma_{2}\le\dots$ are the imaginary parts of the non-trivial zeros of $\zeta(s)$.
We suppose that
$$ \mathop{\sum_{\gamma_{l},\gamma_{k} \le T}}\limits_{|\gamma_{l}-\gamma_{k}|<{c\over \log T}}1\,\ll\,T\log T, \quad T\to\infty, $$
with a certain constant $c>0$. This estimate is a weak form of the Montgomery pair correlation conjecture [1].
Let $D=\bigl\{s\in \mathbb{C}: \tfrac{1}{2}<\sigma<1\bigr\}$, $\mathcal{K}$ be the class of compact subsets of $D$ with connected complements, and let $H_{0}(K)$, $K\in \mathcal{K}$, denote the class of continuous non-vanishing functions on $K$ which are analytic in the interior of $K$. Then we have
Theorem. Suppose that the weak Montgomery conjecture is true. Let $K\in \mathcal{K}$ and $f(s)\in H_{0}(K)$. Then, for every $\varepsilon>0$ and $h>0$,
$$ \liminf_{N\to\infty} \frac{1}{N} \# \left\{ 1\leqslant k\leqslant N: \sup_{s\in K} |\zeta(s+i\gamma_k h)-f(s)|<\varepsilon\right\}>0. $$

In the report, the approximation of analytic functions by $F(\zeta(s+i\gamma_{k}h))$ for some classes of operators $F$ also will be discussed.
[1] H.L. Montgomery, The pair correlation of zeros of the zeta function. In: Analytic Number Theory, (St. Louis Univ., 1972), H.G. Diamond (ed.), Proc. Sympos. Pure Math., Vol. XXIV, Amer. Math. Soc. Providence, 1973. P. 181 – 193.