New correction theorems in the light of a weighted Littlewood--Paley--Rubio de Francia inequality

We prove the following correction theorem: every function $f$ on the circumference $\mathbb{T}$ that is bounded by the ${\alpha }_1$-weight $w$ (this means that $M{w}^2\le C{w}^2$) can be modified on a set $e$ with $\underset{e}{\int }w\le \text{eps}$ so that its quadratic function built up from arbitary sequence of nonintersecting intervals in $\mathbb{Z}$ will not exceed $C\log \frac{1}{\text{eps}}w$.