Hi,

My name is Adam Sheffer and more details about me can be found in my homepage. See also the NYC Discrete Geometry group.

In this blog I write about Discrete Geometry, Additive Combinatorics, and other related topics. One of my main interests is using algebraic techniques to solve combinatorial problems. I also run an REU and am interested in undergraduate (and high-school) math research mentoring.

As you may have noticed, once in a while I cannot resist the urge to write very silly posts.

1. 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:

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

2. 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 $n$ points in the plane spans the smallest number of distinct distances (i.e., out of the $\binom{n}{2}$ distances spanned by pairs of points in the set). For example, by taking $n$ points evenly spaced on a line, you get only $n-1$ distinct distances.

It is known that many $\sqrt{n}\times\sqrt{n}$ lattices span about $n/\sqrt{\log n}$ 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,