I wonder if the various lattice-point theorems, such as Pick's Theorem or Minkowski's Lattice Theorem, have been generalized to the collection of points with rational coordinates no more than height $h$?

A rational $a/b$ in lowest terms has height $\max( |a|,|b| )$. For example, here are the positive rationals of height $\le 5$: $$\left\{\frac{1}{5},\frac{1}{4}, \frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac {2}{3},\frac{3}{4},\frac{4} {5},1,\frac{5}{4},\frac{4}{ 3},\frac{3}{2},\frac{5}{3}, 2,\frac{5}{2},3,4,5 \right\}$$ (Incidentally, the length of this list is given by integer sequence A018805.)

Let us call rationals of height $\le h$, $h$-rationals, and points with both coordinates $h$-rationals, $h$-rational points (my own terminology). Generalizations from lattice points to $h$-rational points seems natural, and likely has been explored...? If so, I would appreciate a reference! That's my primary question.

Here is a specific, Pick's Theorem -like question:

Can the number $i$ of $h$-rational points inside a polygon $P$ be expressed in a form that is not tantamount to enumerating each interior point? Assume we are given the vertex coordinates of $P$, and perhaps the number of boundary points on each edge.

If $P$ is an axis-aligned rectangle, then the number of interior points $i$ can be computed from the number of points $b_x$ lying on the bottom side and the number lying on the left side $b_y$: $i = (b_x - 2)(b_y - 2)$. So, for the rectangle with lowerleft corner $(1,1)$ below, $b_x=8$ and $b_y=7$ leads to $i=30$. But just knowing the total number of boundary points $b=26$, a la Pick's Theorem, is inadequate to determine $i$. It seems feasible that there is a nonlinear transformation that maps the $h$-rational points to the integer lattice, allowing Pick's Theorem to be applied there. Answering this question when $P$ is a triangle should lead to a result for arbitrary $P$ via triangulation.

  • $\begingroup$ Why do you not include negative rational numbers or $0$ when you illustrate the count on the number of rational points with height at most 5? $\endgroup$ – KConrad Jul 1 '13 at 12:55
  • $\begingroup$ @KConrad: Just an arbitrary decision, matching A018805. $\endgroup$ – Joseph O'Rourke Jul 1 '13 at 13:01
  • $\begingroup$ But the count of all rational numbers with height at most $n$ is not that online sequence. $\endgroup$ – KConrad Jul 1 '13 at 13:05
  • $\begingroup$ @KConrad: You are right. I corrected the sentence to say the positive rational numbers of height at most $h$. Thanks. $\endgroup$ – Joseph O'Rourke Jul 1 '13 at 13:15
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    $\begingroup$ I doubt that there can be any formula nearly as nice as Pick's here. You're asking for primitive integer points in a 3-dim. polyhedron (intersection of the cube $|x|\leq H, |y|\leq H, |z|\leq H$ with the polyhedral cone $\lbrace x,y,z \in {\bf R}^3 : (x/z,y/z) \in P \rbrace$). Even without the primitivity condition, the formulas are somewhat complicated, even when as here some of the polyhedron's sides are parallel to the coordinate axes; and the primitive subset surely cannot be any easier to enumerate when the asymptotic behavior as $H\rightarrow\infty$ includes a factor $1/\zeta(3)$ ! $\endgroup$ – Noam D. Elkies Jul 1 '13 at 23:08

One can at least give an asymptotic behaviour of this number, when the height bound grows to infinity.

As shown by Emmanuel Peyre in his paper, Hauteurs et mesures de Tamagawa sur les variétés de Fano (Duke Math. J, 79), there are cases of algebraic varieties over a number field where one can prove an equidistribution statement for the points of bounded height, and the projective space is such a case.

Here is a consequence of that theorem, for the case of the projective line $\mathbf P^1_{\mathbf Q}$. Since you normalized the height of a point $p/q\in\mathbf P^1(\mathbf Q)$ to be $\max(|p|,|q|)$, when written in lower terms, let us define a measure on $\mathbf P^1(\mathbf R)=\mathbf R\cup\{\infty\}$ by $\mu=\mathrm dt/\max(1,|t|)^2$. (For an explanation of the exponent $2$ in the definition of the measure $\mu$, observe that the differential form $\mathrm dt$ has a pole of order $2$ at infinity.)

For any subset $U$ of $\mathbf P^1(\mathbf R)$, let $N(U;B)$ be the number of points of $\mathbf P^1(\mathbf Q)$ of height $\leq B$ which belong to $U$. Peyre's Theorem asserts that for any open subset $U$ of $\mathbf P^1(\mathbf R)$ whose boundary has $\mu$-measure $0$, the following limit formula holds: $$ \frac{N(U;B)}{N(\mathbf P^1(\mathbf R);B)} \to \frac{\mu(U)}{\mu(\mathbf P^1(\mathbf R))} $$ when $B\to\infty$.

The actual statement holds at an adelic level, and for more general normalization of the height, for any number field, as well as for more general varieties.


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