Distinct Distances: Open Problems and Current Bounds

Sheffer, Adam

doi:10.48550/arxiv.1406.1949

Cited by 14 publications

(17 citation statements)

References 36 publications

(66 reference statements)

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“…[42], [26], [34], [37]. For more relatives of the distinct distance problem over the reals we recommend the survey of Sheffer [39].…”

Section: Introductionmentioning

confidence: 99%

On the Pinned Distances Problem in Positive Characteristic

Murphy,

Petridis,

Pham

et al. 2020

Preprint

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We study the Erdős distinct distance conjecture in the plane over an arbitrary field F, proving that any set A, with |A| ≤ char(F) 4/3 in positive characteristic, either determines ≫ |A| 2/3 distinct pair-wise non-zero distances from some point of A to its other points, or the set A lies on an isotropic line. We also establish for the special case of the prime residue field Fp, that the condition |A| ≥ p 5/4 suffices for A to determine a positive proportion of the feasible p distances. This significantly improves prior results on the problem.

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“…[42], [26], [34], [37]. For more relatives of the distinct distance problem over the reals we recommend the survey of Sheffer [39].…”

Section: Introductionmentioning

confidence: 99%

On the Pinned Distances Problem in Positive Characteristic

Murphy,

Petridis,

Pham

et al. 2020

Preprint

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“…Combining the Guth-Katz bound with the induction of Solymosi-Vu, one may get better lower bounds for higher dimensional Euclidean spaces. For example when d = 3, it gives the lower bound N 3/5− for any > 0, see Sheffer [27] for details. There is also a continuous analogue of the problem in geometric measure theory, the Falconer's conjecture, asking about the lower bound of Hausdorff dimension of the sets in R d for which the difference set has positive Lebesgue measure.…”

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confidence: 99%

Erdős distinct distances in hyperbolic surfaces

Lu¹,

Meng²

2020

Preprint

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In this paper, we study the number of distinct distances among any N points in hyperbolic surfaces. For Y from a large class of hyperbolic surfaces, we show that any N points in Y determines ≥ c(Y )N/ log N distinct distances where c(Y ) is some constant depending only on Y . In particular, for Y being modular surface or standard regular of genus g ≥ 2, we evaluate c(Y ) explicitly. We also derive new sum-product type estimates.

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“…For example, the problem has been studied in higher dimensions [23], in finite fields [3,12,16,20], and with bipartite distances [18]. For more information, see this book about the problem [7] and a survey of open distinct distances problems [21].…”

Section: Introductionmentioning

confidence: 99%

Distance problems for planar hypercomplex numbers

FitzPatrick

2020

Preprint

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We study the unit distance and distinct distances problems over the planar hypercomplex numbers: the dual numbers D and the double numbers S. We show that the distinct distances problem in S 2 behaves similarly to the original problem in R 2 . The other three problems behave rather differently from their real analogs. We study those three problems by introducing various notions of multiplicity of a point set.Our analysis is based on studying the geometry of the dual plane and of the double plane. We also rely on classical results from discrete geometry, such as the Szemerédi-Trotter theorem.

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Distinct Distances: Open Problems and Current Bounds

Abstract: We survey the variants of Erdős' distinct distances problem and the current best bounds for each of those.

Cited by 14 publications

References 36 publications

On the Pinned Distances Problem in Positive Characteristic

On the Pinned Distances Problem in Positive Characteristic

Erdős distinct distances in hyperbolic surfaces

Distance problems for planar hypercomplex numbers

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