Hi,

My name is Adam Sheffer and more details about me can be found

in my homepage.
In this blog I write about discrete geometry in general, and mainly about the algebraic techniques that are recently being used in it (the most famous example is probably

the novel proof of Guth and Katz for the distinct distances problem). I am trying to clearly explain the techniques, mention recent developments, open questions, etc.

I also try to have a silly post here and there.

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Hello, I have read some of your thoughts about distinct distances. I have been exploring distances in the lattice in relation to the categorization of space-filling curves, which naturally tile in the lattice. I have several questions relating to the squared distances between points in the square and triangular lattices. The following is where my questions began:

http://archive.org/stream/BrainfillingCurves-AFractalBestiary/BrainFilling#page/n37/mode/2up

I am currently trying to understand some of the properties of these squared distances and how they relate to prime and composite numbers.

I would love to hear any thoughts you might have on this subject.

Thanks!

-Jeffrey Ventrella

Dear Jeffrey,

Thank you for writing. I’m afraid that my knowledge in fractals is practically nonexistent. I have been studying distances in lattices from a more combinatorial viewpoint. Specifically, there is a longstanding problem asking what set of points in the plane spans the smallest number of distinct distances (i.e., out of the distances spanned by pairs of points in the set). For example, by taking points evenly spaced on a line, you get only distinct distances.

It is known that many lattices span about distinct distances, and this is conjectured to be tight. More specifically, Paul Erdős conjectured that the smallest number of distinct distances is obtained by the triangular lattice.

If there’s any connection between the two I’ll be happy to hear about it.

Best,

Adam