MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR Yu. V. LINNIK, B. F. SKUBENKO

ON THE ASYMPTOTICS OF INTEGER MATRICES OF THE THIRD ORDER

Questions of the ergodic theory of integer matrices, developed by the authors in papers \((^1,^2)\), require, in particular, the solution of the following problem. Consider integer square matrices (i.e., matrices) \(X\) with prescribed determinant \(N\). Consider the surface \(\det(X)=N\) and single out on it some “reasonably defined” domain \(\Omega\). For large \(N\) it is required to specify the asymptotics for the number of integer matrices lying in the domain \(\Omega\).

This problem was solved for matrices of the second order in \((^3)\). In the present note we prove its solution for matrices of the third order. Apparently, the corresponding generalization of the method indicated here will lead to a solution for matrices of arbitrary order.

Let a domain \(\Omega\) be given among unimodular matrices

\[ A=\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

with boundary consisting of a fixed number of smooth surfaces and with the condition \(|a_{ij}|<K\) (\(K\) is an arbitrarily large constant). Further, let \(N>5\) be any natural number. Introduce the substitution \(x_{ij}=a_{ij}N^{1/3}\). We have, obviously,

\[ N=\begin{vmatrix} x_{11} & x_{12} & x_{13}\\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33} \end{vmatrix}. \tag{1} \]

Denote by \(f(\Omega,N)\) the number of integer primitive solutions of equation (1) with the condition \(A\in\Omega\), \(x_{ij}=a_{ij}\cdot N^{1/3}\), \(A=(a_{ij})\).

Theorem.

\[ f(\Omega,N)\sim G(N)\frac{\operatorname{mes}(\Omega)}{\zeta(2)\zeta(3)} \]

for fixed \(\Omega\) and as \(N\to\infty\), where

\[ G(N)=\prod_{p_i^{n_i}} \frac{(p_i^{\,n_i+2}-1)(p_i^{\,n_i+1}-1)-(p_i^{\,n_i-1}-1)(p_i^{\,n_i-2}-1)} {(p_i-1)^2(p_i+1)}; \]

\[ N=p_1^{n_1}p_2^{n_2}\cdots p_i^{n_i}\cdots p_k^{n_k} \]

is the canonical decomposition of \(N\); \(\operatorname{mes}(\Omega)\) is the Haar measure on the group of unimodular matrices.

This theorem follows directly from a lemma, which is of independent interest.

Lemma. For any given integers \(a_{11},a_{12},a_{21},a_{22}\) satisfying \(|a_{11}|\), \(|a_{12}|\), \(|a_{21}|\), \(|a_{22}|<N^{1/3}\), i.e. \((a_{11},a_{12},a_{21},a_{22})=1\),

\[ \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix} =q < N^{2/3}, \]

there exist primitive solutions of the equation

\[ \begin{vmatrix} a_{11} & a_{12} & v_1\\ a_{21} & a_{22} & u_1\\ v_2 & u_2 & t \end{vmatrix}=N. \tag{2} \]

in the region

\[ n_i < v_i \le m_i,\qquad k_i < u_i \le l_i, \]

\[ |n_i| \asymp |m_i| \asymp |k_i| \asymp |l_i| \asymp N^{1/3}, \]

\[ m_1-n_1=\Delta_1,\qquad m_2-n_2=\Delta_2,\qquad l_1-k_1=\Delta_3,\qquad l_2-k_2=\Delta_4,\qquad \Delta_j \asymp N^{1/3}, \]

and the number of such solutions is expressed by the formula

\[ W(\Delta_1,\Delta_2,\Delta_3,\Delta_4,q,N) = \prod_{j=1}^{4}\Delta_j\,\frac{\varphi(q)}{q^2} \left(1+O\left(q^{-1/13}\right)\right), \]

where

\[ q=\left| \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right|. \]

From this lemma and Lemma 15 of the paper \((^3)\), the present theorem is derived. Our lemma is proved with the aid of the known lemma of I. M. Vinogradov on Fourier series \((^4)\) and the following observation.

Let

\[ \left| \begin{array}{ccc} a_{11} & a_{12} & a_2\\ a_{21} & a_{22} & b_2\\ a_1 & b_1 & \sigma \end{array} \right|=N \]

be an arbitrary integral primitive solution of equation (2) with the condition g.c.d. \((a_{11},q)=1\); then all other solutions of equation (2) will have the form (in self-explanatory notation)

\[ \left| \begin{array}{ccc} a_{11} & a_{12} & a_2z' + y a_{11}\\ a_{21} & a_{22} & b_2z' + y a_{21}\\ a_1z + x a_{11} & b_1z + x a_{12} & t \end{array} \right|=N, \]

where \(zz'\equiv 1 \pmod q\), and \(t\) is determined uniquely by \(N\) and all the other components.

To prove the lemma it is necessary to estimate from above the sum

\[ \sum_{\substack{l_1,l_2,l_3,l_4=-\infty\\ |l_1|+|l_2|+|l_3|+|l_4|\ne 0}}^{+\infty} C_{l_1l_2l_3l_4} \sum_{\substack{x,y,z\;(\mathrm{mod}\ q)\\(z,q)=1}} \exp\left[ \frac{2\pi i}{q} \left\{ z(a_1l_1+b_2l_2)+z'(a_2l_3+b_2l_4)+ \right.\right. \]

\[ \left.\left. +x(a_{11}l_1+a_{12}l_2)+y(a_{11}l_3+a_{12}l_4) \right\} \right]; \]

\[ \text{1) }\ |C_{l_1l_2l_3l_4}| \le \sum_{j=1}^{4}\Delta_j q^{-4}; \qquad \text{2) }\ |C_{l_1l_2l_3l_4}| \le \frac{1}{(\lambda_1\lambda_2\lambda_3\lambda_4)^r}, \]

where

\[ \lambda_j \ge \max\left(\Delta_j^{-1}q,\ \frac{1}{r}\left|\Delta_j l_j q^{-1}q^{-1/2+\varepsilon}\right|\right) \]

(\(\varepsilon\) arbitrarily small).

The written sum is estimated as \(O(q^{3-1/12+\varepsilon_1})\) (\(\varepsilon_1\) is an arbitrarily small positive quantity) with the aid of the known estimates of André Weil \((^5)\).

Received
27 VI 1962

CITED LITERATURE

\(^1\) Yu. V. Linnik, Tr. III Vsesoyuzn. matem. syezda, 3, 21 (1958).
\(^2\) B. F. Skubenko, DAN, 135, No. 4, 794 (1960).
\(^3\) Yu. V. Linnik, Vestn. LGU, No. 5, 3 (1955).
\(^4\) I. M. Vinogradov, Izbr. tr., 1952.
\(^5\) A. Weil, Proc. Nat. Acad. USA, 34, 204 (1948).