Quantitative aspect of correction theorems

Let $U$ be the Banach space of all functions $g$ on the unite circle $\mathbb T$, such that the series $\sum_{j\ge0}\hat{g}(j)z^j$, $\sum_{j<0}\hat{g}(j)z^j$ converge uniformly on $\mathbb T$. Supply $U$ with the norm $\|\cdot\|_U$,
$$ \|g\|_U\overset{\text{def}}=\sup\{|\sum_{m\le j<n} \hat{g}(j)\xi^j|:m,n\in\mathbb Z,n\in\mathbb Z,|\xi|=1\}. $$
We prove the following quantitative refinement of the classical D. E. Mensov's theorem: if $f\in L^\infty(\mathbb T)$ and $0<\varepsilon<1$ then there exists a function $g$ in $U$ such that $\operatorname{mes}\{f\ne g\}\le C_\varepsilon$ $\|g\|_U\le C(\log1/{\varepsilon})\|f\|_\infty$, $C$ an absolute constant. We also show that the multiplier $\log1/\varepsilon$ is the best possible and describ a general scheme of proving theorems similar to that stated above for some other pairs ($L^\infty(S\mu),X$) (instead of ($L^\infty(\mathbb T),U$). For this scheme to be applicable it is sufficient to assume, for example, that certain “weak type 1-1 inequality” holds for elements of the spaces $X^*$. Such an inequality does hold if $X=U$ (this fact was proved by S. A. Vinogradov by means of Carleson–Hunt almost everywhere convergence theorem), but the scheme appears to be useful in some other cases as well.