An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice - MathOverflow

An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

2012-10-20T20:25:17Z

In a previous question: (http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a lattice point in an $A_2$ hexagonal lattice. The user emiliocba noted that such an approximation was provided by:

Lax, P.D., Phillips, R.S. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46(3), pp. 280 – 350 (1982).

With a best current error term of $O(r^{\frac{2}{3}})$ provided by:

Levitan, B.M. Asymptotic formulae for the number of lattice points in Euclidean and Lobachevskii spaces. Russian Mathematical Surveys 42(3), pp. 13 - 42 (1987).

My question is now if there exists an exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in an $A_2$ hexagonal lattice. We know that an exact counting solution exists for the $Z^2$ integer lattice using the Floor[] function (see - http://mathworld.wolfram.com/GausssCircleProblem.html ):

$N(r) = 1 + 4*Floor[r] + 4*\sum^{Floor[r]}_{i=1} Floor[(r^2-i^2)^{\frac{1}{2}}]$

Can we write a similar counting function for the $A_2$ lattice?

For a list of example values, the number of lattice points in a circle of diameter $r$ (i.e. radius $\frac{r}{2}$) centered on a lattice point in an $A_2$ hexagonal lattice, is given in (http://oeis.org/A053416/list).

For $n = {0, 1, 2, ...}$ we have $N(r) = {1, 1, 7, 7, 19, 19, 37, 43, 61, 73, 91}$.

Answer by Will Sawin for An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

2012-10-20T20:52:56Z

Assume the lattice is generated by vectors $(1,0)$ and $(1/2,\sqrt{3}/2$.

Then the number of lattice points in the column with $x$ coordinate $k/2$ is $1+2 * Floor[ ((4r^2-k^2)/3)^{1/2}]$ if $k$ is even and $2*Floor[ ((4r^2-k^2)/3)^{1/2}+1/2]$ if $k$ is odd.

So we are going to write two sums, one for $k=2i$ and $i=1$ to $Floor[r]$ and one for $k=2i+1$ and $i=0$ to $Floor[r-1/2]$, of these rows.

Then we're going to add the column at $0$. This gives the exact counting formula

$1+2*Floor[r]+2*Floor[r/ \sqrt{3}]+4* \sum_{i=1}^{Floor[r]} Floor[ \sqrt{(4r^2-4i^2)/3}]$

$+4*\sum_{i=0}^{Floor[r-1/2]} Floor[ \sqrt{(4r^2-(2i+1)^2)/3}+1/2]$

I didn't double-check these calculations so there might be some mistakes, but it's clear that some version of this formula is correct.

Answer by Yoav Kallus for An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

2012-10-20T22:09:14Z

This Mathematica formula reproduces the numbers in the OEIS and should be self explanatory:

n[r_] := Sum[ 1 + 2 Floor[Sqrt[r^2 - 3 x^2]], {x, -Floor[r/Sqrt[3]], Floor[r/Sqrt[3]]}] + Sum[2 Floor[ Sqrt[r^2 - 3 x^2] + 1/2], {x, -Floor[(r/Sqrt[3]) + 1/2] + 1/2, Floor[(r/Sqrt[3]) + 1/2] - 1/2}]

http://oeis.org/A053416 has the values corresponding to n[1/2], n[1], n[3/2], n[2] etc.

Assuming the lattice is generated by $(0,1)$ and $(\sqrt{3}/2,1/2)$, then the first sum counts the number of points of the form $(\sqrt{3}x,y)$ where $x,y\in\mathbb{Z}$, $\sqrt{3}x\le r$, and $3x^2+y^2\le r^2$. The second sum counts the number of points of the form $(\sqrt{3}x,y)$ where $x,y\in(\mathbb{Z}+1/2)$, $\sqrt{3}x\le r$, and $3x^2+y^2\le r^2$.