Subsequences of the Fourier sums of functions with a given modulus of continuity

It is proved that for each modulus of continuity $\omega(\delta)$ in the class $H_\omega$ there exists a function $f$ such that for any increasing sequence $\{n_i\}_{i=1}^\infty$ of natural numbers there is a point $x$ at which
\begin{gather*} \varlimsup_{t\to\infty}\frac{S_{n_i}(f,x)-f(x)}{\omega(n_i^{-1})\log{n_i}}\geqslant A>0,\\ \varliminf_{t\to\infty}\frac{S_{n_i}(f,x)-f(x)}{\omega (n_i^{-1})\log{n_i}} \leqslant-A<0, \end{gather*}
where $A$ is an absolute constant. Also considered is the approximation by sequences of Fourier sums of functions of bounded variation with given modulus of continuity.
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