Quantitative aspect of correction theorems. II

Let $0<\varepsilon\le1$, $F\in C(\mathbb T)$, $E=\{F\ne0\}$, $\delta>0$. Then there exists a function $G$ with uniformly convergent Fourier series such that $|G|+|F-G|\le(1+\delta)|F|$, $m\{F\ne G\}\le\varepsilon mE$ and $\sup\{|\sum_{k\le j\le l}\hat G(j)\zeta^j|\colon\zeta\in\mathbb T,\ k\le l\}\le\mathrm{const}\|F\|_\infty(1+\log\varepsilon^{-1})$. Bibliography: 3 titles.