On uniform boundedness of a rational distance set in the plane

Makhul, Mehdi; Shaffaf, Jafar

doi:10.1016/j.crma.2012.01.010

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…The idea of using uniformity to study rational distance sets first appeared in the paper of Makhul and Shaffaf [12] who used uniformity for curves [5]. To obtain our Theorem 1.1 (which generalizes [12]), we use both uniformity for curves [5] as well as uniformity for surfaces [8].…”

Section: Methods Of Proofmentioning

confidence: 99%

“…We then show that Lang's conjecture implies uniform bounds for these sets of rational points, using results of Caporaso–Harris–Mazur [5] and Hassett [8]. The idea of using uniformity to study rational distance sets first appeared in the paper of Makhul and Shaffaf [12] who used uniformity for curves [5]. To obtain our Theorem 1.1 (which generalizes [12]), we use both uniformity for curves [5] as well as uniformity for surfaces [8].…”

Section: Introductionmentioning

confidence: 99%

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The Erdős–Ulam problem, Lang's conjecture and uniformity

Ascher

Braune

Turchet

2020

Bull. London Math. Soc.

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A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.

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Section: Methods Of Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Erdős–Ulam problem, Lang's conjecture and uniformity

Ascher

Braune

Turchet

2020

Bull. London Math. Soc.

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“…In the same circle of ideas, the weak Lang conjecture was used [12] to show that if S is a rational distance set of R 2 that intersects any line in only finitely many points, then there is a uniform bound on the cardinality of the intersection of S with any line. Recently, Ascher et al [2] considered rational distance sets S ⊂ R 2 such that no line contains all but at most four points of S, and no circle contains all but at most three points of S. They showed by assuming the weak Lang conjecture that there exists a uniform bound on the cardinality of such sets S.…”

Section: Introductionmentioning

confidence: 99%

On the Number of Perfect Triangles with a Fixed Angle

Makhul

2020

Discrete Comput Geom

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Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle 0 < θ < π, the number of perfect triangles with an angle θ is finite.

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“…We then show that Lang's Conjecture implies uniform bounds for these sets of rational points, using results of Caporaso-Harris-Mazur [CHM97] and Hassett [Has96]. The idea of using uniformity to study rational distance sets first appeared in the paper of Makhul and Shaffaf [MS12], who used uniformity for curves [CHM97]. To obtain our Theorem 1.1 (which generalizes [MS12]), we use both uniformity for curves [CHM97] as well as uniformity for surfaces [Has96].…”

Section: Introductionmentioning

confidence: 99%

“…The idea of using uniformity to study rational distance sets first appeared in the paper of Makhul and Shaffaf [MS12], who used uniformity for curves [CHM97]. To obtain our Theorem 1.1 (which generalizes [MS12]), we use both uniformity for curves [CHM97] as well as uniformity for surfaces [Has96].…”

Section: Introductionmentioning

confidence: 99%

The Erdős-Ulam problem, Lang's conjecture, and uniformity

Ascher,

Braune,

Turchet

2019

Preprint

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A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.

show abstract

On uniform boundedness of a rational distance set in the plane

Cited by 6 publications

References 11 publications

The Erdős–Ulam problem, Lang's conjecture and uniformity

The Erdős–Ulam problem, Lang's conjecture and uniformity

On the Number of Perfect Triangles with a Fixed Angle

The Erdős-Ulam problem, Lang's conjecture, and uniformity

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