The Reverse Littlewood--Offord problem of Erdős
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910765846364160 |
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| author | He, Xiaoyu Juskevicius, Tomas Narayanan, Bhargav Spiro, Sam |
| author_facet | He, Xiaoyu Juskevicius, Tomas Narayanan, Bhargav Spiro, Sam |
| contents | Let $ε_{1},\ldots,ε_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\ldots,v_n\in \mathbb{R}^2$,
$$\Pr\left[||ε_1 v_1+\ldots+ε_n v_n||_2 \leq \sqrt{2}\right]\geq \frac{c}{n}.$$
This resolves the only remaining conjecture from the seminal paper of Erdős on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Reverse Littlewood--Offord problem of Erdős He, Xiaoyu Juskevicius, Tomas Narayanan, Bhargav Spiro, Sam Probability Combinatorics Let $ε_{1},\ldots,ε_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\ldots,v_n\in \mathbb{R}^2$, $$\Pr\left[||ε_1 v_1+\ldots+ε_n v_n||_2 \leq \sqrt{2}\right]\geq \frac{c}{n}.$$ This resolves the only remaining conjecture from the seminal paper of Erdős on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions. |
| title | The Reverse Littlewood--Offord problem of Erdős |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2408.11034 |