On the divergence of Lagrange interpolation processes on sets of the second category

If $\omega$ is a real nondecreasing semiadditive function, continuous on $[0;1]$, such that $\omega(0)=0$ and $\varlimsup_{n\to\infty}\omega\bigl(\frac1n\bigr)\ln n>0$, then for every matrix of interpolation knots on $[0;1]$ there are a function $f$, continuous on $[0;1]$, whose modulus of continuity $\omega(f,\delta)=O\{\omega(\delta)\}$, and a set $\mathscr E$ of second category on $[0;1]$ such that the Lagrange interpolation process for $f$ diverges everywhere on $\mathscr E$.
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