Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets

We study lower bounds for dyadic square functions of indicator functions. In the case of the dyadic square function $S_{2}$ we obtain a sharp lower bound: for every measurable $A \subset {[0,1)}$, we have $$\|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrt{\tau}\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}},$$ where $\tau$ is the first exit time from $(0,1)$ of a standard Brownian motion started at $|A|$, and $|A|^{*}:=\min\{|A|,1-|A|\}$. This estimate gives logarithmic improvement over the classical Burkholder--Davis--Gundy lower bound $|A|^{*}$. In addition, we show a sharp inequality $$\|S_{1}(\mathbbm{1}_{A})\|_{1} \ge T(|A|)\asymp |A|^{*}\log_{2}\frac{1}{|A|^{*}},$$ where $T(x)=\sum_{k=0}^{\infty}\frac{\operatorname{dist}(2^{k}x,\mathbb{Z})}{2^{k}}$ is the Takagi function.