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arXiv 2510.05044math.CO
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带奇偶性条件的反向Littlewood--Offord问题

Reverse Littlewood--Offord problems with parity conditions

Lawrence Hollom, Gregory B. Sorkin

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13 pages
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AI总结：

该研究探讨带奇偶性条件的反向Littlewood--Offord问题，证明高维下随机带符号和落在原点附近的概率下确界对n的奇偶性敏感，并给出二维奇数n时概率指数衰减的构造。

摘要：

我们研究随机带符号和$\xi_1 v_1 + \dotsb + \xi_n v_n$落在距原点给定距离$r$内的概率，其中$v_1,\dotsc,v_n \in \mathbb{R}^d$是固定单位向量，$\xi_1,\dotsc,\xi_n$是独立均匀分布在$\{-1,+1\}$上的随机变量。特别地，我们的结果表明，对于某些$r$值，该概率的下确界对$n$的奇偶性非常敏感。\n 我们证明，对任意$d\geq 3$，存在某个$\varepsilon = \varepsilon(d) > 0$，使得对任意满足$n \not\equiv d \pmod{2}$的$n$以及$\mathbb{R}^d$中的单位向量$v_1,\dotsc,v_n$，存在符号$\eta_1,\dotsc,\eta_n \in \{-1,+1\}$满足$\\|\sum_{i=1}^n \eta_i v_i\\| \leq \sqrt{d - \varepsilon}$，因此$\mathbb{P}(\\| \xi_1 v_1 + \dotsb + \xi_n v_n \\| \leq \sqrt{d-\varepsilon}) > 0$。这与$n\equiv d \pmod{2}$的情况形成对比，后者中上述概率可以为零。当$d=2$且$n$为奇数时已有更多结论，在这种情况下我们给出一种构造，证明$\mathbb{P}(\\|\xi_1 v_1 + \dotsb + \xi_n v_n\\| \leq 1)$可随$n$增大呈指数衰减。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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