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On the average number of best approximations of linear forms. (English. Russian summary) Zbl 1295.11078
Izv. Math. 78, No. 2, 268-292 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 2, 61-86 (2014).
The author considers the following generalization of best approximations in the multi-dimensional case: let \(f:\mathbb R^n\rightarrow [0,\infty)\) be a continuous piecewise differentiable ray function, that is, for any \(x\in\mathbb R^n\) and \(\lambda\in\mathbb R\), we have \(f(\lambda x)=|\lambda| f(x)\) and \((0\leq f(x), f(x)=0 )\Leftrightarrow x=0\). A non-zero vector \((u,v)\in \mathbb Z^n\times \mathbb Z\) is referred to as a \(f\)-best approximation of a linear form \(L:\mathbb R^n\rightarrow \mathbb R\) if there exists no non-zero vector \((u',v')\in \mathbb Z^n\times\mathbb Z\) such that \(|Lu'-v'|\leq |Lu-v|, \;f(u')\leq f(u)\), where at least one of the two inequalities is strict.
Let \(\mathfrak B_f(\alpha)\) be the set of \(f\)-best approximations of a linear form \(x\in \mathbb R^n\rightarrow \alpha_1x_1+\dots+\alpha_nx_n\). The set \(\mathfrak B_f(\alpha)\) is finite iff the numbers \(\alpha_1,\dots,\alpha_n,1\) are linearly dependent over \(\mathbb Z\). For every \(P\geq 1\) and \(\alpha\in[0,1)^n\) the author considers the set \(\mathfrak B_f(\alpha,P)=\{ (u,v)\in\mathfrak B_f(\alpha)\;:\;f(u)\leq P \}\). This set is finite and \(\#\mathfrak B_f(\alpha,P)\ll_f \ln P+1\).
For every real \(R>1\) the author defines a set \(\Delta_n(R)\) consisting of the rational vectors \(\alpha=( \alpha_1,\dots,\alpha_n)\) with \(\alpha_i=\frac{P_i}{Q}\), where \(P_i, Q\) are integers such that \(0\leq P_i<Q\leq R, \;i=1,\dots,n\). If \(\alpha\in \Delta_n(R)\) Minkowski’s convex body theorem shows that for every best approximation \((u,v)\in \mathfrak B_f(\alpha)\) with \(v\not=\alpha_1u_1+\dots +\alpha_n u_n\) we have \(f(u)\leq C_f^* R^{1/n}\) where the constant \(C_f^*\) depends only on \(f\). Let \[ E_f(R,P)=\frac{1}{\#\Delta_n(R)}\sum_{\alpha\in \Delta_n(R)}\#\mathfrak B_f(\alpha,P) \] and \[ E_f(R)=\frac{1}{\#\Delta_n(R)}\sum_{\alpha\in \Delta_n(R)}\# \mathfrak B_f(\alpha) \] be the average numbers of \(f\)-best approximations of linear forms with rational coefficients in \(\Delta_n(R)\). Let \(\mathcal E_f(P)=\int_{[0,1)^n}\#\mathfrak Bf(\alpha,P)d\alpha\) be the expectation of the number of \(f\)-best approximations \((u,v)\) for \(f(u)\leq P\). The Lebesgue integral \(\mathcal E_f(P)\) exists and is finite. In the one dimensional case (that is, for \(n=1\) and \(f(x)=|x|)\), it has been shown (see the references in [A. V. Ustinov, Applications of Kloosterman sums to arithmetic and geometry (Russian). LAMBERT Academic Publishing (2011; Zbl 1322.11004); G. Lochs, Monatsh. Math. 65, 27–52 (1961; Zbl 0097.03602); H. Heilbronn, Abh. Zahlentheorie Anal., zur Erinnerung an E. Landau, 87–96 (1968; Zbl 0212.06503); J. W. Porter, Mathematika 22, 20–28 (1975; Zbl 0316.10019)] ) that \(E_f(R)=C_1\ln R+\mathcal O(1),\) \(\mathcal E_f(P)=C_1\ln P+\mathcal O(1)\) with \(C_1=\frac{24\ln 2}{\pi^2}\). The author generalizes to the multidimensional case:
The formulae \[ \begin{aligned} & E_f(R,P)=C_f\ln P+\mathcal O_f(1), \quad P\leq C_f^*R^{1/n},\tag{1}\\ & \mathcal E_f(P)=C_f\ln P+\mathcal O_f(1),\tag{2} \end{aligned} \] hold for every \(n\geq 1\) and any continuous piecewise-differentiable ray function \(f: \mathbb R^n\rightarrow\mathbb R\) where \(C_f\) is a positive constant depending only on \(f\) made explicit by the author in the article.
The proof of (1) uses the ideas in [A. A. Illarionov, St. Petersbg. Math. J. 24, No. 2, 301–312 (2013); translation from Algebra Anal. 24, No. 2, 154–170 (2012; Zbl 1273.11103)], where an asymptotic formula for the average value of the cylindrical minima of integer lattices is obtained. The approach in [Zbl 1273.11103] enables to reduce the evaluation of the sum \(\sum_{\alpha\in\Delta_n(R)}\#\mathfrak B_f(\alpha,P)\) to finding the number of \((n+1)(n+1)\) integer matrices \(M\) such that \[ M\in\Omega_f,\;|\det M |\in[1,R], \;d_n(M)=1, \;f(m_{11},\dots, m_{n1}\leq P), \] where \(\Omega_f\) is a subset of \(GL_{n+1}(\mathbb R)\) and \(d_n(M)\) is the greatest common divisor of the cofactors of the elements of the last row of \(M\).
11J13 Simultaneous homogeneous approximation, linear forms
11J17 Approximation by numbers from a fixed field
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