Estimates of the integral modulus of continuity of functions with rarely changing Fourier coefficients

The functions under consideration are those satisfying the condition $\Delta a_i=\Delta b_i=0$ for all $i\ne n_j$, where $\{n_j\}$ is a lacunary sequence.
An asymptotic estimate of the rate of decrease of the modulus of continuity in the $L$-metric of such functions in terms of their Fourier coefficients is obtained.