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 an $\alpha_1$-weight $w$ (this means that $Mw^2\le Cw^2$) can be modified on a set $e$ with $\int_ew<\varepsilon$ so that the quadratic function built up from $f$ with the help of an arbitary sequence of nonintersecting intervals in $\mathbb Z$ will not exceed $C\log(\frac1\varepsilon)w$.