Average Number of Lattice Points in a Disk

Sujay Jayakar & Bob Strichartz

Introduction

In this paper, we investigated the problem of estimating the difference between the area of a disk and the number of lattice points contained within the disk. Directly providing tight estimates for the difference is a long standing problem, but we instead look at different averages. We examined the problem through the lens of counting eigenvalues of the Laplacian on different tori, Klein bottles, and projective planes. A preprint draft of the paper is available here.

Table of Contents

  1. Plots from the paper
  2. Circle plots
    1. Counting and differences
    2. Convergence
    3. Frequency plots
  3. Ellipse plots
    1. Counting and differences
    2. Frequency plots
  4. Torus and Klein bottle plots
  5. Numerics

I. Plots from the paper

Click on any of the images for a PDF version. All of the plots in this section use grids with 0.1 spacing, and computations involving g are summed over the disk of radius 100.
Figure 1: D(t)
Figure 1 gives us the difference D(t) between the number of lattice points contained within the disk of radius √t/2π and its area. See Section II.A for more circle plots of this nature.
Figure 2: t-1/4D(t)
The difference D(t) is conjectured to be O(t1/4+ε) for every ε>0. If this conjecture holds, then Figure 2 will be to(1).
Figure 3: A(t)
Figure 3 is a plot of the integral of Figure 2.
Figure 4: t1/4A(t)
Figure 4 is Figure 3 multiplied by t1/4. Since we showed that A(t) is O(t-1/4), Figure 4 is bounded asymptotically.
Figure 5: g(√t)
In the paper we describe a mean value zero, almost periodic function g such that g(√t) is a good estimate for t1/4A(t). See Figure 5 for g(√t).
Figure 6: t1/4A(t) - g(√t)
The error of our estimate is given in Figure 6.
Figure 7: √t(t1/4A(t) - g(√t))
In particular, the error is O(t-1/2), so the plot is bounded asymptotically after multiplying by √t.
Figure 8: ~A(t)
Here, we instead compute D((2πr)2), which yields the difference for the disk of radius r, and integrate it to get Figure 8.
Figure 9: A(2πt2)/(2√(2π))
In the paper we show that ~A has the same asymptotics as A(2πt2)/(2√(2π)), which is plotted in Figure 9.
Figure 10: A(2πt2)/(2√(2π)) - ~A(t)
The difference between Figures 8 and 9 is given in Figure 10.
Figure 11: g(√t), a1= 2, a2= 1/2.
Figures 11 and 12 are plots of g(√t) and its error for a particular family of ellipses (namely, those with a1= 2 and a2= 1/2). See Section III.A for more plots in this vein.
Figure 12: t1/4A(t) - g(√t), a1= 2, a2= 1/2.

II. Circle plots

A. Counting and differences

Here we include data for g, t1/4A(t), and their difference that goes out to 1,000,000 instead of just 10,000.
The equivalent of Figure 4 on a much larger domain
Figure 5 for t going up to 1,000,000
Figure 6 on the larger domain

B. Convergence

In this section, we investigate the convergence of our approximations to A and g. We approximated the integral in A by sampling the integrand over a partition of the domain and averaging the values. As we make the partition finer, the Riemann sums converge to the integral. The convergence plots for A in both the sup and l1 norm are given below. Similar plots for g are given below. Instead of altering the grid, we evalute the infinite sum given in equation (1.14) over domains of different sizes.

C. Frequency plots

To see the distribution of D(t), we computed D(t) over the interval [0, 100000] and binned the data into bins of size 0.1 to make a histogram. The results are depicted below.

III. Ellipse plots

A. Counting and differences

Here, we reproduce Figures 4, 5, and 6 with different families of ellipses. The case where a1 = a2 = 1 is depicted below. Click on each image to see the plots for different ellipses.
Figure 4 for different ellipses
Figure 5 for different ellipses
Figure 6 for different ellipses

B. Frequency plots

Here we repeat the binning process from Section II.C for different ellipses.
Figure 6 for different ellipses

IV. Torus and Klein bottle plots

The above figure is a plot of the differences for the counting functions of the torus and Klein bottle. Here, we multiply the difference by t1/4. Finally, this plot bins the difference to form a histogram.

V. Numerics

Along the way, we ran many numerical computations to test our hypotheses. The majority of them are written in Python, with a few in C for performance. All of our source code is available for download here.