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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 $$M$$.
MSC:
 11J13 Simultaneous homogeneous approximation, linear forms 11J17 Approximation by numbers from a fixed field
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