What is recorded here?
Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
The result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.
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People and systems
Larry Guth; Nets Hawk Katz
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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English catalogue narrative
Guth and Katz: a near-linear lower bound for planar distinct distances
Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.
Scientific scope
The result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.
Reading status
This is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.
People and systems
Larry Guth; Nets Hawk Katz
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.