On the Erdős distinct distances problem in the plane

By Larry Guth, Nets Hawk Katz. 2015

432 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eac1a39f3962b3e44fb3088f1ab16c66520142a4

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In Applied Mathematics

In Computer Graphics and Computer-Aided Design

In Geometry and Topology

Citations in the corpus

An L2-identity and pinned distance problem (2018, background): “The idea of using group action argument to attack distance problem dates back to the solution to the Erdős distance problem ([8], [12]).”
New Bounds on the Dimensions of Planar Distance Sets (2018, background): “Katz [8] (building up on work of Gy.”
A semi-algebraic version of Zarankiewicz's problem (2014, methodology): “We combine Theorem 5.1 with Hilbert polynomials to obtain a variant of the polynomial ham sandwich theorem (for the original theorem, see for example [20]).”
Improved Fourier restriction estimates in higher dimensions (2018, methodology): “This method was introduced by Guth and Katz in their resolution of the Erdős distance conjecture [14] and was first applied to the restriction problem by Guth in [12] (the latter work also incorporated a refinement to the original partitioning method of [14] due to Solymosi and Tao [29]).”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, background): “|, which was at the mercy of the Cauchy-Schwarz inequality) by Guth and Katz [13] who showed that |∆(P )| ≫”
Sharp estimates for oscillatory integral operators via polynomial partitioning (2017, methodology): “A key tool in the present analysis of Hörmander-type operators is the following polynomial partitioning result, which is a variant of the polynomial partitioning theorem introduced in [15], and is based on the classical polynomial ham-sandwich theorem of Stone and Tukey [29].”
Restriction estimates using polynomial partitioning II (2016, methodology): “erse complete intersection. 5.2. Polynomial partitioning. Polynomial partitioning is a key tool in our arguments. Our presentation here is a minor variation on the polynomial partitioning result from [GK]. We begin by stating a partitioning result from [G]: Theorem 5.2. (Theorem 1.4 in [G]) Suppose ”
Upper bounds for $s$-distance sets and equiangular lines (2016, background): “For the former case, the most well-studied problem is the famous Erdős distinct distances problem for the plane [27] which was essentially solved by Guth and Katz [36] who proved that the size of an s-distance set on the plane is O(s ln s).”