Uniform Convergence of Translation Operators

We denote N, R, C the sets of natural, real and complex numbers respectively. Let (λn), n ∈ N be an unbounded sequence numbers. Costakis has proved following result. There exists entire function f with property: for every x, y R 0 , θ ∈(0,1) a there is subsequence natural (mn), such that, compact subset L ⊆ In present paper we show that constant cannot replaced by any non-constant G. This so even if one demands convergence in (*) only single radius r positive number θ. result related problem existence common universal vectors uncountable family sequences translation operators.