Geodesic
Quantitative aspect of correction theorems. II
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 83-91 
S. V. Kislyakov. Quantitative aspect of correction theorems. II. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 83-91. http://geodesic.mathdoc.fr/item/ZNSL_1994_217_a7/
@article{ZNSL_1994_217_a7,
     author = {S. V. Kislyakov},
     title = {Quantitative aspect of correction {theorems.~II}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--91},
     year = {1994},
     volume = {217},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1994_217_a7/}
}
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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.