Reverse Littlewood--Offord problems with parity conditions
Lawrence Hollom, Gregory B. Sorkin
TL;DR
This work studies parity-sensitive Littlewood–Offord-type questions for random signed sums in $\mathbb{R}^d$. The authors constructively demonstrate extreme behavior in 2D and, for $d\ge 3$, establish a balancing phenomenon: when $n \not\equiv d \pmod 2$, there exist deterministic signs producing a near-origin sum $\|\sum_{i=1}^n \eta_i v_i\|\le\sqrt{d-\varepsilon}$, implying a positive probability of being within $\sqrt{d-\varepsilon}$; whereas parity alignment can force the corresponding probability to vanish. The core methodology combines convex-geometric tools (S(V), Z(V), approximating properties), a constructive exponential-small-probability example in 2D, and a stability/dichotomy framework to prove the $(d-\varepsilon)$-balancing bound for $d\ge 3$. These results illuminate how parity interacts with geometric structure to influence anti-concentration and raise natural open questions about optimal constants and phase transitions in higher dimensions.
Abstract
We consider the probability that the random signed sum $ξ_1 v_1 + \dotsb + ξ_n v_n$ lies within a given distance $r$ of the origin, where $v_1,\dotsc,v_n \in \mathbb{R}^d$ are fixed unit vectors and $ξ_1,\dotsc,ξ_n$ 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 $η_1,\dotsc,η_n \in \{-1,+1\}$ such that $\|\sum_{i=1}^n η_i v_i\| \leq \sqrt{d - \varepsilon}$, and so $\mathbb{P}(\| ξ_1 v_1 + \dotsb + ξ_n v_n \| \leq \sqrt{d-\varepsilon}) > 0$. 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 $\mathbb{P}(\|ξ_1 v_1 + \dotsb + ξ_n v_n\| \leq 1)$ can decay exponentially as $n$ increases.