Reverse Littlewood--Offord problems with parity conditions

We consider the probability that the random signed sum $\xi_1 v_1 + \dotsb + \xi_n v_n$ lies within a given distance $r$ of the origin, where $v_1,\dotsc,v_n \in \mathbb{R}^d$ are fixed unit vectors and $\xi_1,\dotsc,\xi_n$ are independently and uniformly distributed on $\{-1,+1\}$. In particular, our results demonstrate that, for certain values of $r$, the infimum of this probability is very sensitive to the parity of $n$. We prove that, for any $d\geq 3$, there is some $\varepsilon = \varepsilon(d) > 0$ such that for any $n \not\equiv d \mod 2$ and unit vectors $v_1,\dotsc,v_n\in \mathbb{R}^d$, there are signs $\eta_1,\dotsc,\eta_n \in \{-1,+1\}$ such that $\|\sum_{i=1}^n \eta_i v_i\| \leq \sqrt{d - \varepsilon}$, and so $\mathbb{P}(\| \xi_1 v_1 + \dotsb + \xi_n v_n \| \leq \sqrt{d-\varepsilon}) > 0$. This is in contrast to the case of $n\equiv d \mod 2$, wherein the above probability can be zero. More is known if $d=2$ and $n$ is odd, and in this case we present a construction demonstrating that $\mathbb{P}(\|\xi_1 v_1 + \dotsb + \xi_n v_n\| \leq 1)$ can decay exponentially as $n$ increases.