On the asymptotic distribution of integer matrices
Let \(s, N\geq 2\) be integers, \(M_s(\mathbb Z), M_s(\mathbb R)\) square matrices of order \(s\), \(M_s(\mathbb Z,N):=\{X\in M_s(\mathbb Z)\;|\;\det X=N\}\), \(TM_s(\mathbb Z):=\{((t_{ij}))\in M_s(\mathbb Z)\;|\;t_{ij}=0\) for \(j>i,\;0\leq t_{ij} < t_{ii}\) for \(j\leq i,\;i=1,\dots,s\}\), \(TM_s(\mathbb Z,N):=TM_s(\mathbb Z)\cap M_s(\mathbb Z,N)\), \(\mathrm{GL}_s(\mathbb R):=\{X\in M_s(\mathbb R) \;|\;\det X\not=0\}\). Let the set \(\Omega\subset \mathrm{GL}_s(\mathbb R)\) satisfying the two conditions:NEWLINENEWLINE(A) \(\Omega=\{((x_{ij}))\;|\;(x_{i1},\dots,x_{is})\in V_i,\;i=1,\dots,s\}\), where \(V_i, i=1,\dots,s\) is a connected cone in \(\mathbb R^s\) with the vertex at the point \(x=0\), the boundary of \(V_i\) is piecewise differentiable (an hypersurface \(S\subset \mathbb R^s\) is \textit{piecewise differentiable} if \(S\) consists of a finite number of differentiable hypersurfaces).NEWLINENEWLINE(B) There exists a positive constant \(C\) such that \(\prod_{i=1}^s\max_{1\leq j\leq s}|x_{ij}|\leq C\cdot\det X\) for any \(X=((x_{ij}))\in\Omega\).NEWLINENEWLINEThe main aim of the article is to obtain an asymptotic formula for the number of integer matrices \(M\in \Omega\) with \(\det M=N\). NEWLINELet \(\mathcal D_s(\mathbb R_+)\) be the set of diagonal matrices \(((x_{ij}))\in \mathrm{GL}_s(\mathbb R)\) such that \(x_{ii}\in \mathbb R^+,\;i=1,\dots,s\). Consider the \(s(s-1)\)-dimension manifold \(\mathrm{PGL}_s(\mathbb R)=\mathcal D_s(\mathbb R^+)\backslash \mathrm{GL}_s(\mathbb R)\) (the set of right cosets). Let \(\mathcal P(\Omega)\) be the image of \(\Omega\in \mathrm{GL}_s(\mathbb R)\) under the projection \(\mathrm{GL}_s(\mathbb R)\rightarrow \mathrm{PGL}_s(\mathbb R)\).NEWLINENEWLINEThe author defines a certain measure \(\mu\) on \(\mathrm{PGL}_s(\mathbb R)\): suppose that \(k=\{k_1,\dots,k_s\}\) is a permutation of \((1,\dots,s)\) and \(\theta=(\theta_1,\dots,\theta_s), \;\theta_i=\pm 1\). Let \(\mathrm{GL}_s(\mathbb R,k,\theta):=\{((x_{ij}))\in \mathrm{GL}_s(\mathbb R) \;|\;x_{i k_i}=\theta_i,\;i=1,\dots,s\}\). Let \(\mathrm{PGL}_s(\mathbb R,k,\theta)=\mathcal P(\mathrm{GL}_s(\mathbb R,k,\theta)\) be a map on \(\mathrm{PGL}_s(\mathbb R)\).NEWLINEDefine a measure \(\mu\) on the map \(\mathrm{PGL}_s(\mathbb R,k,\theta)\) as \(\mu(w)=\int_W\frac{dW(X)}{|\det X|^s}\) for \(w\in \mathrm{PGL}_s(\mathbb R,k,\theta),\) where \(W\) is the prototype of \(w\) under the projection \(\mathrm{GL}_s(\mathbb R,k,\theta)\rightarrow \mathrm{PGL}_s(\mathbb R,k,\theta)\) and \(dW(X)\) is a differential of an \(s(s-1)\)-dimensional Lebesgue measure on \(\mathrm{GL}_s(\mathbb R,k,\theta)\subset \mathbb R^{s(s-1)}\) taken at some point \(X\). It can be proved that the measure \(\mu\) so defined does not depend on the choice of the map and is so well defined on \(\mathrm{PGL}_s(\mathbb R)\). The author proves:NEWLINENEWLINESuppose that the set \(\Omega\subset \mathrm{GL}_s(\mathbb R) \) satisfies the conditions (A) and (B). For any integer \(N\geq 2\), the number of integer matrices \(M\) such that \(M\in \Omega\) and \(\det M = N\) is equal to NEWLINE\[NEWLINE\frac{\mathcal R_s(N)}{\zeta(2)\dots\zeta(s)}\cdot \Big(\frac{\mu(\mathcal P(\Omega))}{(s-1)!}\ln^{s-1} N+\mathcal O(\chi(N)\cdot \ln^{s-2} N)\Big),NEWLINE\]NEWLINE where \(\mathcal R_s(N)=\# TM_s(\mathbb Z,N)\) ( \(\#\) number of elements of a set), \(\zeta\) is the Riemann function and \(\chi(N)=1+\sum_{p|N}\frac{\ln p}{p}\) with \(p\) primes.NEWLINENEWLINEThe case \(s=2\) can be proved straightforwardly with the methods of [\textit{H. Heilbronn}, Abh. Zahlentheorie Anal., 87--96 (1968; Zbl 0212.06503)], the case \(s=3\) was considered in [the author, ``The average value of local minima of three-dimensional integer lattices with a given determinant'', Vladivostok: Dal'nauka (2010). (Preprint/FEB RAS. Khabarovsk division of the institute for Applied Mathematics, N. 02 (in Russian)].
scientific article; zbMATH DE number 6077916
scientific article