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Bardwell-Evans, Sam; Sheffer, Adam

doi:10.1016/j.jcta.2019.02.010

Cited by 3 publications

(3 citation statements)

References 15 publications

(49 reference statements)

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Mentioning

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Order By: Relevance

“…For part (a), see for example [2,Section 11.3]. For part (b), see for example [1,Section 7.1]. We say that a function f : R d → R is smooth if all its partial derivative of all orders exist.…”

Section: Paper Structurementioning

confidence: 99%

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Distinct Distances in $R^3$ Between Quadratic and Orthogonal Curves

Aldape¹,

Liu²,

Pylypovych³

et al. 2023

Preprint

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We study the minimum number of distinct distances between point sets on two curves in R 3 . Assume that one curve contains m points and the other n points. Our main results:(a) When the curves are conic sections, we characterize all cases where the number of distances is O(m + n). This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is Ω(min{m 2/3 n 2/3 , m 2 , n 2 }).(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is O(m + n). This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is Ω(min{m 2/3 n 2/3 , m 2 , n 2 }).

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Section: Paper Structurementioning

confidence: 99%

“…Equivalently, we may switch the roles of v 1 and v 2 in (19). After this switch, we compute coeff [1] + coeff [t 10 ] = 2c 6 y 6 1 z 2 1 z 2 2 . Since this expression equals zero and c, y 1 = 0, we have that z 1 = 0 or z 2 = 0.…”

Section: A Mathematica Computations: the Case Of Two Ellipsesmentioning

confidence: 99%

Distinct Distances in $R^3$ Between Quadratic and Orthogonal Curves

Aldape¹,

Liu²,

Pylypovych³

et al. 2023

Preprint

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“…. , q k ∈ Q in that order leads to k edges of the form (q i , q i+1 , C) in Step (2), where q k+1 = q 1 . Thus, for each point p ∈ P, we constructed n edges in the graph.…”

Section: Theorem 9 ([27]mentioning

confidence: 99%

On Bipartite Distinct Distances in the Plane

Mathialagan

2021

Electron. J. Combin.

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Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

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