On one spectrum of universality for Walsh system

In the present work it is shown that the set $D=\left\{\displaystyle\sum_{i=0}^{\infty}\delta_i2^{N_i} :\delta_i=0,1\right\}$ for every sequence $N_0<N_1\ldots<N_i\ldots$ of natural numbers can be changed into the set of the form $\Lambda=\left\{k+o(\omega(k)):k\in D\right\}$ , where $\omega(k)$ is an arbitrary, tending to infinity at $k\to+\infty$ sequence, such that $\Lambda$ is the spectrum of universality for Walsh system.