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On an application of Guth-Katz theorem

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Alex Iosevich, Oliver Roche-Newton, Misha Rudnev

Combinatorics · Submitted 7 Mar 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products.

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arXiv ID1103.1354 (version 5)
Pith IDZNVLAFLZ, Pith's permanent record of this paper and its review
CitationAlex Iosevich, Misha Rudnev, Oliver Roche-Newton. "On an application of Guth-Katz theorem." arXiv:1103.1354 (2011). https://pith.science/paper/ZNVLAFLZ
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,471 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. This in particular implies a sum-product bound $$ |A\cdot A\pm A\cdot A|\geq c\frac{|A|^2}{\log |A|} $$ for a discrete $A \subset {\mathbb R}$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. result 0.93 0.96 0.87 asserted 1.00
s1 This in particular implies a sum-product bound $$ |A\cdot A\pm A\cdot A|\geq c\frac{|A|^2}{\log |A|} $$ for a discrete $A \subset {\mathbb R}$. interpretation 0.88 0.85 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v5
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. This in particular implies a sum-product bound $$ |A\cdot A\pm A\cdot A|\geq c\frac{|A|^2}{\log |A|} $$ for a discrete $A \subset {\mathbb R}$.

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2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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