Extending Erdős–Beck's theorem to higher dimensions

Do, Thao

doi:10.1016/j.comgeo.2020.101625

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…The many variations of this theorem constitute a rich field of study [3,7,5,13]. These results also find applications in other domains from group theory [16] to the Kakeya problem in harmonic analysis [6].…”

Section: Our Family Of Constructionsmentioning

confidence: 82%

Sharp Szemerédi-Trotter Constructions in the Plane

Guth¹,

Silier²

2021

Preprint

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Section: Our Family Of Constructionsmentioning

confidence: 82%

Sharp Szemerédi-Trotter Constructions in the Plane

Guth¹,

Silier²

2021

Preprint

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“…The right generalization of Beck's Theorem to rank-k flats was found by Do [4], using Lund's notion of "essential dimension" [8]. A similar notion had previously been introduced by Lovász [6, Theorem 2.3] who, together with Yemini, found applications to bar-and-joint rigidity problems [7].…”

Section: Bounding the Number Of Rank-k Flatsmentioning

confidence: 89%

On the complex-representable excluded minors for real-representability

Campbell

Geelen

2020

Journal of Combinatorial Theory, Series B

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“…Beck's theorem can be stated as follows. Beck's theorem has important applications in different areas of mathematics, and it has opened a new research field in combinatorial geometry, see for instance [1], [4], [6], [7], [13], [16]. Another important family of problems in combinatorial geometry is to bound the number of curves with a given degree that are determined by A and satisfy other conditions (for example, in the Sylvester-Gallai type results, the curves have to pass through few points of A), see for instance [2], [3], [5], [19].…”

Section: Introductionmentioning

confidence: 99%

Beck's Theorem for Plane Curves

Huicochea¹

2020

Electron. J. Combin.

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Let $d\in\mathbb{Z}^+t$, $\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\mathbb{K}^2$. Denote by $\mathcal{C}_{d,\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\mathbb{K}^2$ and $\mathcal{C}_{\leq d,\mathbb{K}}:=\bigcup_{e=1}^d\mathcal{C}_{e,\mathbb{K}}$. For any $C_1\in \mathcal{C}_{d,\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\in\mathcal{C}{d,\mathbb{K}}$ such that $C_2\cap A\supseteq C_1\cap A$, we have that $C_1=C_2$; we denote by $\mathcal{D}_{d,\mathbb{K}}(A)$ the family of elements of $\mathcal{C}_{d,\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\mathbb{K}=\mathbb{R}$ and $A$ is not collinear, then $$|\mathcal{D}_{1,\mathbb{R}}(A)|=\Theta\left(|A|\min_{C\in \mathcal{C}_{1,\mathbb{R}}}|A\setminus C|\right).$$ In this paper we generalize Beck's theorem showing that for all $d\in\mathbb{Z}^+$, there exists a constant $c=c(d)>0$ such that if $\min_{C\in\mathcal{C}_{\leq d,\mathbb{K}}}|A\setminus C|>c,$ then $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$

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Extending Erdős–Beck's theorem to higher dimensions

Cited by 4 publications

References 17 publications

Sharp Szemerédi-Trotter Constructions in the Plane

Sharp Szemerédi-Trotter Constructions in the Plane

On the complex-representable excluded minors for real-representability

Beck's Theorem for Plane Curves

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