Things are moving fast!

Lately exciting new results concerning distinct distances are constantly being discovered. Marcos Charalambides just uploaded this paper to arXiv. The main result of the paper is the following.

Theorem 1 (Charalambides). Consider a curve \gamma \subset {\mathbb R}^d  of degree m  , and let \cal P  be a set of n  points that are contained in \gamma  . If no irreducible component of \gamma  is an algebraic helix, then the number of distinct distances that are determined by \cal P  is \Omega(n^{5/4})  (where the constant of proportionality depends on d  and m  ).

Algebraic helices are lines and geodesics of the Clifford torus, which is the surface that is parameterized as

(u_1,\cdots,u_k)\to(\alpha_1\cos u_1,\alpha_1\sin u_1,\cdots,\alpha_k\cos u_k,\alpha_k\sin u_k)\subset{\mathbb R}^{2k},

where 2k\le d  and \alpha_1,\cdots,\alpha_k >0  are constants. One reason why I find this result quite interesting is its connection to characterizing sets that determine a small number of distinct distances (say, o(n)  distinct distances; see a brief discussion in this post). Only a few properties of such sets are known. Theorem 1 implies that if \cal P  is a planar n  -point set that determines o(n)  distinct distances, then no constant degree curve, except possibly for planar algebraic helices, can contain \Omega(n^{4/5})  points of \cal P  .

Charalambides states that “the proof relies on a link to a certain structural rigidity question on curves”. I still need time to digest how this proof works, but I am always excited to see new connections between combinatorial geometry problems and other parts of math!

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2 thoughts on “Things are moving fast!

  1. Janos Pach and Frank de Zeeuw just posted a paper to the arXiv (http://arxiv.org/abs/1308.0177) which gives a better exponent of 4/3 (instead of the 5/4 in the article mentioned above) for the case when the ambient dimension is 2, by considering the same kind of implicitly-defined algebraic curves as in Sharir-Sheffer-Solymosi and solving the resulting algebraic problem.

    Their algebraic proof doesn’t seem to generalize as is to higher ambient dimensions, but perhaps a combination of this new result with some analytic techniques (which do feature rather prominently in the article above) can give the improved exponent of 4/3 for all ambient dimensions.

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