The optimal point-concentration bound for definable sets

Let $S\subseteq\mathbb{R}^d$ be a set definable in an o-minimal structure and containing no line segment. For nonzero vectors $a_1,\dots,a_n\in\mathbb{R}^d$, let $\xi_1,\dots,\xi_n$ be i.i.d. Rademacher random variables and set

$$X=a_1\xi_1+\dots+a_n\xi_n.$$

Optimal point-concentration conjecture. In the setting of Theorem baby-forwards, one has $\Pr(X\in S)\le C_S/\sqrt n$ for some constant $C_S$ depending only on $S$.

The preceding theorem proves the weaker bound $n^{-1/2+\alpha}$ for every $\alpha>0$ when $n$ is sufficiently large. The conjecture asks for the optimal Erdős--Littlewood--Offord order of magnitude in this o-minimal setting.