A note on N. Bary's one conjecture (Q1637543)

A trigonometric sine series \(\sum\limits_{n=1}^\infty b_n\sin nx\) with special properties is constructed. One of the properties, the possibility of which is proved by \textit{V. Ya. Kozlov} [Mat. Sb., Nov. Ser. 26(68), 351--364 (1950; Zbl 0039.29302)], is that the sequence of its partial sums \(S_m(x)\) contains a subsequence \(S_{m_k}(x)\) such that it converges to zero everywhere as \(k\to\infty\) and this convergence is uniform on \([\delta,\pi-\delta]\) for every \(\delta\in(0,\pi/2)\). Bary conjectured in 1960 that for such a series, the subsequence \(m_k\) possesses the property \(\frac{m_{k+1}}{m_k}\to\infty\) as \(k\to\infty\). The author disproved this conjecture in 2003. In the paper under review, the author strengthens his result.