Uniform approximation of partial sums of a Dirichlet series by shorter sums and $\Phi$-widths

It is shown that each Dirichlet polynomial $P$ of degree $N$ which is bounded in a certain natural Euclidean norm, admits a nontrivial uniform approximation on the corresponding interval on the real axis by a Dirichlet polynomial with spectrum containing significantly fewer than $N$ elements. Moreover, this spectrum is independent of $P$.
Bibliography: 19 titles.