On a radial projection conjecture and pinned directions in finite spaces Article Swipe
Paige Bright
,
Ben Lund
,
Thang Pham
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2311.05127
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2311.05127
We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over $\mathbb{F}_q$, let $k \in \{1,2,\ldots,d-1\}$, and let $E \subseteq \mathbb{F}_q^d$ be an arbitrary set of points. We prove two results. First, if $q^{k-1} < |E| \leq 100^{-1}q^{k}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $50^{-1}|E|$ points is bounded above by $40q^k$. This establishes a conjecture of Lund, Pham, and Thu. Second, if $30q^{k} \leq |E| \leq q^{k+1}$, then the number of points $y$ such that the projection of $E$ from $y$ contains fewer than $M \leq 4^{-1}q^k$ points is bounded above by $300q^kM|E|^{-1}$. We also have an application to a pinned directions problem. Specifically, if $E\subset \mathbb{F}_q^d$ with $|E| > 30q^k$, then there is a point $y \in E$ such that the set of lines incident to $y$ and at least one other point of $E$ determines $q^k/4$ distinct slopes.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2311.05127
- https://arxiv.org/pdf/2311.05127
- OA Status
- green
- Cited By
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4388585734
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- OpenAlex ID
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https://openalex.org/W4388585734Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2311.05127Digital Object Identifier
- Title
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On a radial projection conjecture and pinned directions in finite spacesWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2023Year of publication
- Publication date
-
2023-11-09Full publication date if available
- Authors
-
Paige Bright, Ben Lund, Thang PhamList of authors in order
- Landing page
-
https://arxiv.org/abs/2311.05127Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2311.05127Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2311.05127Direct OA link when available
- Concepts
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Conjecture, Combinatorics, Projection (relational algebra), Mathematics, Physics, Discrete mathematics, AlgorithmTop concepts (fields/topics) attached by OpenAlex
- Cited by
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1Total citation count in OpenAlex
- Citations by year (recent)
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2024: 1Per-year citation counts (last 5 years)
- Related works (count)
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10Other works algorithmically related by OpenAlex
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