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 of degree , and let be a set of points that are contained in . If no irreducible component of is an*algebraic helix,

*then the number of distinct distances that are determined by is (where the constant of proportionality depends on and ).*

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

where and 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, distinct distances; see a brief discussion in this post). Only a few properties of such sets are known. Theorem 1 implies that if is a planar -point set that determines distinct distances, then no constant degree curve, except possibly for planar algebraic helices, can contain points of .

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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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.

Thanks Marcos!

I’ll definitely read this soon.