Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem

In this paper we prove that $\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }\ge \sqrt{\text{Var}(X)})\ge 7\mathrm{/}32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{ń} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvořák and Klein. Moreover we fully determine the function $f(y)={\inf }_X\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }\ge y\sqrt{\text{Var}(X)})$ where the $\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein.