B. F. SKUBENKO

ASYMPTOTIC DISTRIBUTION AND ERGODIC PROPERTIES OF INTEGER POINTS ON A ONE-SHEETED HYPERBOLOID

(Presented by Academician I. M. Vinogradov on 28 VI 1960)

§ 1. We consider the surface of a one-sheeted hyperboloid (H) in Cartesian coordinates (a, b, c),

[
b^{2}-ac=D,\qquad D>0.
\tag{1}
]

The number (D) is assumed to be an integer, odd, and not a perfect square. On the surface (H) we single out the classical reduction domain (\Delta)

[
b^{2}-ac=D,\qquad 0<b<\sqrt D,\qquad \sqrt D-b<|a|<\sqrt D+b.
\tag{2}
]

By (\Delta_1) we shall denote a finitely connected figure with piecewise smooth boundary, contained in (\Delta). By the symbol (V(\Delta_1)) we shall mean the volume of the cone with base (\Delta_1) and vertex at the origin. By the symbol (A(\Delta_1)) we shall mean the hyperbolic solid angle, i.e. (A(\Delta_1)=V(\Delta_1)/D^{3/2}). By (T(\Delta_1)) we denote the number of integer points satisfying the condition (\gcd(a,2b,c)=1) and lying on the surface (\Delta_1).

Suppose that there exists a prime number (p) such that ((D/p)=+1). Then the following holds.

Theorem 1.

[
T(\Delta_1)=\frac{A(\Delta_1)}{A(\Delta)}\,T(\Delta)\,(1+\eta(p,D));
]

[
\eta(p,D)\to 0 \quad \text{for fixed } p,\ A(\Delta_1)\text{ and as }D\to\infty.
]

Theorem 1 is proved by a method which enabled Yu. V. Linnik, in particular, to prove the analogous theorem for the case of a two-sheeted hyperboloid.

§ 2. The essential difference between the reduction domains of the one-sheeted and two-sheeted hyperboloids does not permit a trivial transfer of the results of ((^1)) to the case of the one-sheeted hyperboloid. In particular, the theorem on cycles turned out to be essential in the proof of Theorem 1. We formulate it in terms of Gauss composition of binary quadratic forms.

Denote by (A_1, A_2, B) classes of forms of determinant (D>0); (\sqrt D) is irrational. Here (A_1\cdot B=A_2) (in the sense of composition). Denote by (n_1, n_2) the number of representatives (in the reduction domain) of quadratic forms corresponding respectively to the classes (A_1, A_2), and let (0<a=\min(B)). Then there holds the

Theorem on cycles 1.

[
\frac{n_1}{n_2}<c\log(1+a),\qquad \text{where }c\text{ is an absolute constant.}
]

By making the reasoning more complicated and applying considerations of a geometric character, one can prove a stronger theorem.

Theorem on cycles 2.

[
\frac{n_1}{n_2}<\mathrm{const}\cdot \log[1+\log(1+a)],
]

or, as a consequence, for (D>3)

[
\frac{n_1}{n_2}<\mathrm{const}\cdot \log\log D.
]

§ 3. Denote by (L=\begin{pmatrix} b&-a\ c&-b\end{pmatrix}) a matrix with zero trace; (a,b,c) are integers; (\gcd(a,2b,c)=1); (b^2-ac=D>0); (D) is nonsquare, (\sqrt D) is irrational. Further, denote by (R) a matrix of the second order with integer components.

Let (\det R=p^k), ((D/p)=+1). Then there exists such a matrix (R), with (\det R=p^k), that (RLR^{-1}=L') will be integral and (\gcd(a',2b',c')=1), and moreover (L'\in\Delta). The (n)-fold rotation of the form (RLR^{-1}) forms an abstract flow. Note that (R) is chosen nonuniquely, and therefore it is essential to specify the direction of the flow.

It is very important that there exists a rule of selection for each (L\in\Delta) of a single matrix (R) from the set of (R_i) for which (R_iLR_i^{-1}=L') ((i=0,1,2,\ldots)), (L'\in\Delta), (L') integral. In this connection we shall denote an operation of the type (RLR^{-1}) by (\mathcal L), and the image (L') by (\mathcal LL); the (n)-fold operation will be denoted by (\mathcal L^{(n)}), and accordingly the image by (\mathcal L^{(n)}L). Further, by (f_{\Delta_1}(L)) we shall denote the characteristic function of the set of primitive integral points. Then the following holds:

Theorem 2 (ergodic).

[
\frac{1}{s}\sum_{n=1}^{s} f_{\Delta_1}\bigl(\mathcal L^{(n-1)}L\bigr)
=
\frac{A(\Delta_1)}{A(\Delta)}
\bigl(1+\eta(p^k,\Delta_1,D)\bigr)
]

for all images of primitive points (L), with the possible exception of (o(T(\Delta))); (s\ge c_0\log D), and (\eta(p^k,\Delta_1,D)\to0) for fixed (p^k,\Delta_1) and (D\to\infty).

Let (\mathfrak M) be any set of primitive integral points lying in the domain (\Delta). We shall denote by (\mathcal L^{(n)}\mathfrak M) the set of primitive integral points of the domain (\Delta) to which these points “flow” after an (n)-fold transformation (\mathcal L); (M(\mathfrak M)) is the number of points of (\mathfrak M); (M(\mathcal L^{(n)}\mathfrak M\cap\Delta_1)) is the number of points of the set of images (\mathcal L^{(n)}\mathfrak M) (counting also overlaps), whose images lie in the set (\Delta_1). Then the following holds:

Theorem 3. Let (n=0,1,2,\ldots), (s\ge c_1\log D), (M(\mathfrak M)>\varepsilon_0T(\Delta)) ((\varepsilon_0>0) is any fixed number). Then for all indices (n), with the possible exception of (s\cdot o(1)) ((D\to\infty)) such indices, we have

[
M\bigl(\mathcal L^{(n)}\mathfrak M\cap\Delta_1\bigr)
=
\frac{A(\Delta_1)}{A(\Delta)}\,M(\mathfrak M)
\bigl(1+\eta(p^k,\Delta_1,\varepsilon_0,D)\bigr),
]

where (\eta(p^k,\Delta_1,\varepsilon_0,D)\to0) for fixed (p^k,\Delta_1,\varepsilon_0) and (D\to\infty).

For all the theorems we take

[
A(\Delta)=\frac{4}{3}(2\log 2-1).
]

It should be noted that, in contrast to the case of the sphere ({}^{(2)}) and the two-sheeted hyperboloid, Theorem 1 (on the asymptotic distribution) does not follow from Theorem 3.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
27 VI 1960

CITED LITERATURE

1. Yu. V. Linnik, Vestn. LGU, No. 2, 3 (1955); No. 5, 3 (1955); No. 8, 15 (1955).
2. Yu. V. Linnik, Matem. sborn., 43 (85), 2, 257 (1957).