LAUSR

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts $$\zeta (s+i\tau )$$ζ(s+iτ), $$\tau \in \mathbb {R}$$τ∈R, of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts $$\zeta (s+ix_kh)$$ζ(s+ixkh), $$k\in \mathbb {N}$$k∈N, $$h>0$$h>0, where $$\{x_k\}\subset \mathbb {R}$${xk}⊂R is such that the sequence $$\{ax_k\}$${axk} with every real $$a\ne 0$$a≠0 is uniformly distributed modulo 1, $$1\le x_k\le k$$1≤xk≤k for all $$k\in \mathbb {N}$$k∈N and, for $$1\le k$$1≤k, $$m\le N$$m≤N, $$k\ne m$$k≠m, the inequality $$|x_k-x_m| \ge y^{-1}_N$$|xk-xm|≥yN-1 holds with $$y_N> 0$$yN>0 satisfying $$y_Nx_N\ll N$$yNxN≪N.