Integer superharmonic matrices have the Apollonian structure

Geometry and Topology 2 Minutes

You need to know: Matrix, matrix multiplication, transpose A^T of matrix A, notation {\mathbb Z}^2 for the set of 2\times 1 matrices x=\begin{bmatrix}x_1\\x_2\end{bmatrix} with integer entries, norm ||x||=\sqrt{x_1^2+x_2^2} of x\in{\mathbb Z}^2, real matrix (matrix with real entries), square matrix, symmetric matrix, trace of a square matrix (sum of its diagonal entries), positive semidefinite matrix, Euclidean plane {\mathbb R}^2, lines and circles in the plane, tangent circles, notation x=x_0 for vertical line through the point (x_0, 0), notation C((x,y),r) for the circle with centre (x,y)\in{\mathbb R}^2 and radius r, small o notation.

Background: For x,y\in{\mathbb Z}^2 we write x\sim y if ||x-y||=1. For a function g:{\mathbb Z}^2\to {\mathbb Z}, let \Delta g(x) = \sum\limits_{y\sim x}(g(y)-g(x)). A 2\times 2 matrix A is called integer superharmonic if there exists a function g:{\mathbb Z}^2\to {\mathbb Z} such that g(x) = \frac{1}{2}x^TAx+o(||x||^2) and \Delta g(x) \leq 1 for all x\in {\mathbb Z}^2. Let S_2 be the set of all 2\times 2 real symmetric matrices.

By general circle on the plane we mean circle or line. Every triple T of pairwise tangent general circles has exactly two general circles (called Soddy general circles) tangent to all three. An Apollonian circle packing (ACP) generated by T is a minimal collection of general circles containing T that is closed under the addition of Soddy general circles. For k\in{\mathbb Z}, let {\cal B}_k be the ACP generated by lines x=2k, x=2k+2 and circle C((2k+1,0),1). Let {\cal B}=\bigcup\limits_{k\in{\mathbb Z}}{\cal B}_k. To each circle C((x,y),r) \in {\cal B} we associate the matrix A_C =\frac{1}{2}\begin{bmatrix}r+x & y \\y & r-x\end{bmatrix}.

The Theorem: On 12th September 2013, Lionel Levine, Wesley Pegden, and Charles Smart submitted to arxiv a paper in which they proved that matrix A\in S_2 is integer superharmonic if and only if either \text{trace}(A)\leq 2 or there exists a circle C\in{\cal B} such that matrix A_C-A is positive semidefinite.

Short context: Integer superharmonic matrices arise from the theory of partial differential equations (PDE), and it was on open question to characterise them. The Theorem achieves this via connection to Apollonian circle packing, which, for non-specialist, looks like a totally unrelated area of mathematics. Such theorems are useful because they allow to apply methods and results from one area to a different one.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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