How many lattice points are there on a circle or a sphere?

Abstract

In this chapter we study the distribution of points with integral coordinates on spheres.

Appendices

Exercises

• 9.1 ($\maltese$) Investigate the error term in the asymptotic formula of Theorem 9.4.

• 9.2 In this exercise we assume the reader is familiar with basic complex analysis.

  • a. Show that for each $s \in \mathbb C$ with $\mathfrak {R}s > 0$, the integral

    $$ \varGamma (s) : = \int _0^\infty t^{s-1} e^{-t} \, dt $$

    is absolutely convergent.

  • b. Show that the for all s with $\mathfrak {R}s > 0$ we have

    $$ \varGamma (s+1) = s \varGamma (s). $$

  • c. Conclude that the function $\varGamma (s)$, originally defined on $\mathfrak {R}s > 0$, has an analytic continuation to a meromorphic function on all of $\mathbb C$ with simple poles at $s = 0, -1, -2, -3, \dots$. Compute the residues at the poles.

  • d. Show that $1/\varGamma (s)$ is entire.

  • e. Show that for each natural number n, $\varGamma (n) = (n-1)!$.

  • f. Show that for all $s_1, s_2$ with $\mathfrak {R}s_1, \mathfrak {R}s_2 > 0$, we have

    $$ \int _0^1 t^{s_1-1}(1-t)^{s_2-1} \, dt = \frac{\varGamma (s_1)\varGamma (s_2)}{\varGamma (s_1 + s_2)}. $$

• 9.3 Find an easy function $f:\mathbb N\rightarrow \mathbb C$ which does not have an average value.

• 9.4 Compute the volume of the sphere of radius R in $\mathbb R^k$.

• 9.5 Compute the diameter of the unit hypercube in $\mathbb R^k$.

• 9.6 Prove Theorem 9.7.

• 9.7 Show that the function $r_k$ for $k > 2$ does not have an average value. Find a continuous function $f: \mathbb R\rightarrow \mathbb R$ with the same average order as $r_k$.

• 9.8 Prove that for a triangle with side lengths a, b, c with area S which is inscribed in a circle of radius R we have

  $$ abc = 4 RS. $$

• 9.9 Show that if a circle of radius r in $\mathbb R^2$ has three points A, B, C such that the distances AB, AC, BC are rational numbers, then r is a rational number.

• 9.10 Show that every circle in $\mathbb R^2$ with rational radius contains infinitely many points every two of which have rational distance.

• 9.11 Justify Equation (9.1).

• 9.12 Show that for all real numbers $\xi$, $|||\xi ||| = |\xi + [\xi ] - [2\xi ]|$.

• 9.13 Show that for all real numbers $\xi , \eta$,

  $$ |||\xi + \eta ||| \le |||\xi ||| + |||\eta |||. $$

• 9.14 Show that for all $\xi \in \mathbb R$ and $n \in \mathbb Z$, $||| n \xi ||| \le |n| \cdot |||\xi |||$.

• 9.15 Show that for all natural numbers n,

  $$ n \cdot ||| n \sqrt{2} ||| \ge 2 \cdot ||| 2 \sqrt{2} ||| = 6- 4 \sqrt{2}. $$

• 9.16 Show that the real numbers $1, \phi _2, \phi _3, \phi _5, \dots$ appearing in the proof of Theorem 9.10 are linearly independent over the rational numbers.

• 9.17 Suppose $\beta$ is a positive integer, and k a natural number. Show that for each choice of $\gamma _1, \dots , \gamma _k$ such that for i, $|\gamma _i| \le \beta$ and $\gamma _i \equiv \beta \ \mathrm {mod}\ 2$, we have

  $$ \sum _{1 \le i < j \le k} |\gamma _i - \gamma _j| \le \ \frac{k^2 - \delta (k)}{2}\beta $$

  where $\delta (k) = {\left\{ \begin{array}{ll} 0 &{} k \text { even}; \\ 1 &{} k \text { odd}.\end{array}\right. }$. Show that equality is attained if

  • a. k even: k / 2 of the $\gamma _i$’s are equal to $\beta$ and the other k / 2 are equal to $-\beta$;

  • b. k odd: $(k+1)/2$ of the $\gamma _i$’s are equal to $\beta$ and the remaining $(k-1)/2$ are equal to $-\beta$.

• 9.18 Prove inequality (9.5).

• 9.19 Show that for every natural number m there are infinitely many circles centered at the origin with precisely m integral points on their perimeters.

• 9.20 Show that for each natural number n, there are infinitely many circles in $\mathbb R^2$ which contain exactly n lattice points.

• 9.21 This problem is about the celebrated theorem of Georg Pick (1859–1942, Theresienstadt Concentration Camp). A simple proof of this theorem appears in [103].

  • a. Suppose T is a triangle in the plane all of whose vertices are lattice points. Let S be the area of the triangle, E the number of lattice points on the edges, and I the number of lattice points inside the triangle. Show that

    $$ S = I + \frac{1}{2} E -1. $$

  • b. Prove Pick’s theorem: Let $\mathsf P$ be a closed non self-intersecting polygon in $\mathbb R^2$ whose vertices are lattice points. Let S be the area, E the number of lattice points on the edges, and I the number of lattice points inside $\mathsf P$. Then we have

    $$ S = I + \frac{1}{2} E -1. $$

• 9.22 ($\maltese$) Investigate Question 9.11.

• 9.23 ($\maltese$) Do you believe Conjecture 9.12?

• 9.24 ($\maltese$) For each natural number n, consider the sphere $S_n$ defined by

  $$ x^2 + y^2 + z^2 = n $$

  in $\mathbb R^3$, and define $S_n(\mathbb Z)$ to be the collection of points on $S_n$ that have integral coordinates. If $(x, y, z) \in S_n(\mathbb Z)$, then

  $$ (\frac{x}{\sqrt{n}}, \frac{y}{\sqrt{n}}, \frac{z}{\sqrt{n}}) \in S_1. $$

  Investigate the distribution of the resulting points on the sphere $S_1$. Experiment with restricting the sequence of n’s, e.g., squares, primes, etc.

Notes

Gauss’ Circle Theorem

In Theorem 9.4 we showed that if we have a circle of radius r, then the number of lattice points inside the circle is $\pi r^2 + O(r)$. There is a famous conjecture [23, Section F1] asserting that the error term in Gauss’ Circle Theorem is $O(r^{1/2+ \epsilon })$ for any $\varepsilon >0$. Richard Guy describes the problem of proving this conjecture as very difficult. The best result in this direction is due to Martin Huxley who around the year 2000 proved that the error is $O(r^{131/208})$ improving his own earlier result of $O(r^{46/73})$. Note that $46/73 - 131/208 = 0.000329...$.