Geometric Littlewood-Offord problems via lattice point counting
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910976271450112 |
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| author | Grebennikov, Alexandr Kwan, Matthew |
| author_facet | Grebennikov, Alexandr Kwan, Matthew |
| contents | Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $ξ_{1},\dots,ξ_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $ξ_{1}a_{1}+\dots+ξ_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a set of points in convex position, an algebraic variety, or a semialgebraic set. In particular, this resolves conjectures of Fox-Kwan-Spink and Kwan-Sauermann.
We also obtain some corollaries for the polynomial Littlewood-Offord problem, for polynomials that have bounded Chow rank (i.e., can be written as a polynomial of a bounded number of linear forms). For example, one of our results confirms a conjecture of Nguyen and Vu in the special case of polynomials with bounded Chow rank: if a bounded-degree polynomial $F\in\mathbb{C}[x_{1},\dots,x_{n}]$ has bounded Chow rank and ''robustly depends on at least $b$ of its variables'', then $\mathbb{P}[F(ξ_{1},\dots,ξ_{n})=0]\le O(1/\sqrt{b})$. We also prove significantly stronger bounds when $F$ is ''robustly irreducible'', towards a conjecture of Costello. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Littlewood-Offord problems via lattice point counting Grebennikov, Alexandr Kwan, Matthew Combinatorics Number Theory Probability Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $ξ_{1},\dots,ξ_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $ξ_{1}a_{1}+\dots+ξ_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a set of points in convex position, an algebraic variety, or a semialgebraic set. In particular, this resolves conjectures of Fox-Kwan-Spink and Kwan-Sauermann. We also obtain some corollaries for the polynomial Littlewood-Offord problem, for polynomials that have bounded Chow rank (i.e., can be written as a polynomial of a bounded number of linear forms). For example, one of our results confirms a conjecture of Nguyen and Vu in the special case of polynomials with bounded Chow rank: if a bounded-degree polynomial $F\in\mathbb{C}[x_{1},\dots,x_{n}]$ has bounded Chow rank and ''robustly depends on at least $b$ of its variables'', then $\mathbb{P}[F(ξ_{1},\dots,ξ_{n})=0]\le O(1/\sqrt{b})$. We also prove significantly stronger bounds when $F$ is ''robustly irreducible'', towards a conjecture of Costello. |
| title | Geometric Littlewood-Offord problems via lattice point counting |
| topic | Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2505.24699 |