G. Babaev

Distribution of Integer Points on Certain Norm Surfaces

(Presented by Academician I. M. Vinogradov on 19 IV 1960)

§ 1. Distribution of integer points on certain ellipses and hyperbolas.

Let (\omega=\omega(a,b)) be the angle between the rays issuing from the origin and making angles (a) and (b) with the abscissa axis, where (0\le a<b\le 2\pi); (\omega=b-a); (d=1,2,3,7,11,19,43,67,163); (\delta=1), if (-d\equiv 2,3 \pmod 4); (\delta=4), if (-d\equiv 1 \pmod 4). Let (m) be an odd natural number,

[
m=p_1^{\beta_1}p_2^{\beta_2}\cdots p_s^{\beta_s}
]

its canonical decomposition into prime factors; (r) the number of all integer points lying on the ellipse

[
x^2+dy^2=\delta m;
\tag{1}
]

(T) the number of integer points on the ellipse (1) that lie inside the angle of aperture (\omega).

Theorem 1. If

[
\left(\frac{-d}{p_t}\right)=+1
]

for (t=1,2,\ldots,s) (where (\left(\frac{-d}{p_t}\right)) is the Legendre symbol) and

[
\Delta=\sqrt{\frac{\ln(p_1\cdots p_s)}{\ln r}}\to 0
\quad \text{as } m\to\infty,
]

then

[
T=\frac{r}{2\pi}\,[\operatorname{arc\,tg}(\sqrt d\,\operatorname{tg} b)-\operatorname{arc\,tg}(\sqrt d\,\operatorname{tg} a)]+O(r\Delta).
]

Proof. We note the main stages of the proof for the simplest case (d=1). The proof for other (d) is analogous.

Let

[
m=x_j^2+y_j^2,\qquad x_j+iy_j=\sqrt m\,\exp(2\pi i\alpha_j),\qquad 0\le \alpha_j<1,
]

where (x_j,y_j) are integers, (j=1,2,\ldots,r). We are interested in those solutions for which

[
0<A\le \alpha_j\le B<1,\qquad A<B,\qquad b-a=2\pi(B-A).
]

Estimate the sum

[
S_m^{(k)}=\sum_{j=1}^{r}\exp(2\pi ik\alpha_j),
]

where (k) is an integer, (k\ne 0).

All integers (x_j+iy_j) of the field (R(i)) ((R) is the field of rational numbers) with norm (m) are represented in the form

[
x_j+iy_j=(i)^\gamma\prod_{t=1}^{s} c_t^{\gamma_t}{c'_t}^{\beta_t-\gamma_t},
]

where (c_t,c'_t) are prime numbers of the field (R(i)), with (c_tc'_t=p_t), (\gamma=0,1,2,3), (\gamma_t=0,1,\ldots,\beta_t). Taking into account that (c_t^k\ne {c'_t}^k) for any (k\ne 0), we obtain

[
\left|S_m^{(k)}\right|\le \min\left(r,\,4(p_1p_2\cdots p_s)^{k/2}\right).
]

Further, applying Lemma 12 from [1], p. 260, we obtain the proof of the theorem.

Let (d) be a natural square-free number, (d>1); (R(\sqrt d)) a one-class field ((R) is the field of rational numbers); (\varepsilon) a unit of norm 1 such that every unit of norm 1 is obtained in the form (\pm \varepsilon^n), where (n) is some integer. We may assume that (\varepsilon>1). Let (\varphi) be the angle between the ray

[
y=\frac{\varepsilon^2-1}{(\varepsilon^2+1)\sqrt d}\,x,\qquad x\geqslant 0,
]

and the axis of abscissas; let (\omega=\omega(a,b)) be the angle between rays issuing from the origin and making angles (a) and (b) with the axis of abscissas, where

[
\omega=b-a,\qquad 0\leqslant a<b<\operatorname{arc\,tg}\frac{\varepsilon^2-1}{(\varepsilon^2+1)\sqrt d}.
]

Let (m) be odd,

[
m=p_1^{\beta_1}p_2^{\beta_2}\cdots p_s^{\beta_s},\qquad \left(\frac{d}{p_t}\right)=+1\quad \text{for } t=1,2,\ldots,s;
]

(r) is the number of integral points on the hyperbola

[
x^2-dy^2=\delta m,
\tag{2}
]

lying inside the angle (\varphi); (T) is the number of integral points on the hyperbola (2) lying inside the angle (\omega), where (\delta=1), if (d\equiv 2,3\pmod 4), and (\delta=4), if (d\equiv 1\pmod 4). Put

[
u=\frac{(1-\sqrt d\,\operatorname{tg} a)(1+\sqrt d\,\operatorname{tg} b)}
{(1-\sqrt d\,\operatorname{tg} b)(1+\sqrt d\,\operatorname{tg} a)},\qquad
\Delta=\sqrt{\frac{\ln(\varepsilon^{3s}p_1\cdots p_s)}
{\ln\left[r(\varepsilon\ln\varepsilon)^{-s}\right]}}.
]

Theorem 2. If (\Delta\to 0) as (m\to\infty), then

[
T=\frac{r\ln u}{2\ln\varepsilon}+O(r\Delta).
]

Proof. We indicate the main stages of the proof. The hyperbola (2) can be written in the form (\xi\eta=m). Under the logarithmic mapping a point ((\xi,\eta)) with (\xi>0,\ \eta>0) passes to the point ((v,w)=(\ln\xi,\ln\eta)) of the plane ((v,w)); the hyperbola (\xi\eta=m) to the straight line (v+w=\ln m); the angle (\omega(a_1,b_1)), where (a_1\ne 0,\ b_1\ne \pi/2), to the strip bounded by the straight lines (w=v+\ln\operatorname{tg}a_1) and (w=v+\ln\operatorname{tg}b_1); the unit (\varepsilon) is represented in the form of the vector (l(\varepsilon)=(\ln\varepsilon,\ln\varepsilon')), where (\varepsilon') is conjugate to (\varepsilon). The angle (\varphi) passes to the strip bounded by the straight lines (w=v) and (w=v-2\ln\varepsilon).

Let (p_t=c_t c_t'), where (c_t,c_t') are prime numbers of the field (R(\sqrt d)),

[
\mu=\left|\prod_{t=1}^{s} c_t^{\gamma_t} c_t'^{\,\beta_t-\gamma_t}\right|,\qquad
\gamma_t=0,1,2,\ldots,\beta_t.
]

It is clear that (N(\mu)=\mu\mu'=m). The number of such (\mu) will be (r).

Estimate the sum

[
S_m^{(k)}=\sum_{\mu}\exp\left{2\pi i k\,\frac{\ln(\mu/\sqrt m)}{\ln\varepsilon}\right},
]

where (k) is an integer, (k\ne 0), and (\mu) runs through the indicated (r) numbers.

Taking into account that (\ln|c_t/c_t'|/\ln\varepsilon) is an irrational number, we obtain

[
\left|S_m^{(k)}\right|\leqslant
\min\left(r,\ \prod_{t=1}^{s}\frac{1}{2\left(\ln|c_t/c_t'|^k/\ln\varepsilon\right)}\right),
]

where ((\ )) denotes the distance to the nearest integer. Estimating this distance from below, we obtain

[
\left|S_m^{(k)}\right|\leqslant
\min\left(r,\ (\varepsilon\ln\varepsilon)^s(\varepsilon^{3s}p_1\cdots p_s)^k\right).
]

Further, applying Lemma 12 of paper ((^1)), p. 260, and passing to the variables (x,y), we obtain Theorem 2.

Remark 1. For fixed (s) and prime numbers (p_1,p_2,\ldots,\ldots,p_s), the application of A. O. Gelfond’s theorem ((^2)) for estimating ((\quad)) from below gives a better remainder term in Theorem 2.

Remark 2. Using the methods of proof of Theorems 1 and 2, one can obtain theorems on the distribution of integer points on a system of nonequivalent ellipses and hyperbolas, but in view of the cumbersomeness of these theorems we do not give them here.

§ 2. Distribution of integer points on certain surfaces of the third order. Let (R(\theta)) be a one-class field of degree 3 over the field of rational numbers, where (\theta) is a nonreal number; (\omega_1,\omega_2,\omega_3) is a basis of the integers of the field (R(\theta)). In three-dimensional space ((\xi_1,\xi_2,\xi_3)) consider the lattice (\Gamma):

[
\xi_1=x_1\operatorname{Re}\omega_1+x_2\operatorname{Re}\omega_2+x_3\operatorname{Re}\omega_3,
]

[
\xi_2=x_1\operatorname{Im}\omega_1+x_2\operatorname{Im}\omega_2+x_3\operatorname{Im}\omega_3,
]

[
\xi_3=x_1\omega_1''+x_2\omega_2''+x_3\omega_3'',
]

where (\omega_i'') are real conjugates of (\omega_i); (x_1,x_2,x_3) are rational integers.

Consider the surface

[
(\xi_1^2+\xi_2^2)|\xi_3|=m,
\tag{3}
]

where (m) is a natural number. It is known that in the field (R(\theta)) there exist infinitely many prime numbers (c) such that (|cc'c''|=p), where (p) is a rational prime number; (c') is the complex conjugate of (c); (c'') is the real conjugate of (c). We assume that (m) consists exclusively of the product of powers of such prime numbers: (m=p_1^{\beta_1}p_2^{\beta_2}\cdots p_s^{\beta_s}). On the surface (3) we distinguish the region (D_m): (\sqrt[3]{m}\leq |\xi_3|\leq \sqrt[3]{m}\,|\varepsilon|^2), where (\varepsilon) is the fundamental unit of the field (R(\theta)). Let (r) denote the number of points of the lattice (\Gamma) that fall in (D_m). We distinguish a part of the region (D_m): (\sqrt[3]{m}\leq \alpha\sqrt[3]{m}\leq |\xi_3|\leq \beta\sqrt[3]{m}\leq \sqrt[3]{m}\,|\varepsilon|^2). By (T) we denote the number of points of the lattice (\Gamma) falling in this part of (D_m). It is not difficult to compute that

[
r=2\prod_{t=1}^{s}\frac{(\beta_t+1)(\beta_t+2)}{2}.
]

Denote

[
\Delta=\sqrt{\frac{\ln\left(|\varepsilon|^{108s}(p_1\ldots p_s)^3\right)}
{\ln\left(r2^{-6s}|\varepsilon|^{-12s}\ln^{-3s}|\varepsilon|\right)}}.
]

Theorem 3. If (\Delta\to 0) as (m\to\infty), then

[
T=\frac{\ln\beta-\ln\alpha}{2\ln|\varepsilon|}\,r+O(r\Delta).
]

The proof of this theorem is analogous to the proof of Theorem 2.

Mathematical Institute
named after V. A. Steklov
of the Academy of Sciences of the USSR

Received
18 IV 1960

CITED LITERATURE

(^1) I. M. Vinogradov, Selected Works, 1952.
(^2) A. O. Gelfond, Transcendental and Algebraic Numbers, 1952, p. 217.