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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.05044 |
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| _version_ | 1866909827359309824 |
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| author | Hollom, Lawrence Sorkin, Gregory B. |
| author_facet | Hollom, Lawrence Sorkin, Gregory B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reverse Littlewood--Offord problems with parity conditions Hollom, Lawrence Sorkin, Gregory B. Combinatorics 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. |
| title | Reverse Littlewood--Offord problems with parity conditions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.05044 |