Diophantine approximation and parametric geometry of numbers
Dirichlet's Theorem says that for \(n>1\), real numbers \(\xi_1,\dots,\xi_{n-1}\) and \(Q\geq 1\), there is a nonzero point \(\mathbf y=(y, y_1,\dots,y_{n-1})\in\mathbb Z^n\) having \(|y|\leq Q, |\xi_i y-y_i|\leq Q^{-1(n-1)}\) for \(i=1,\dots, n-1\). On the other hand let \(\Lambda\subset \mathbb R^n\) be a lattice and \(\mathcal K\) a closed symmetric convex body in \(\mathbb R^n\) with \(vol(\mathcal K)\geq 2^n \det \Lambda\). From Minkowski, \(\mathcal K\) contains a nonzero lattice point. Now, let \(\Lambda :=\Lambda(\xi)=\Lambda(\xi_1,\dots,\xi_{n-1})\) be the lattice of points \((y,\xi_1 y-y_1,\dots,\xi_{n-1} y-y_{n-1})\) where \(\mathbf y=(y,y_1,\dots,y_{n-1})\) runs through \(\mathbb Z^n\). Let \(\mathcal K(Q)\) consists of points \((x_1,x_2,\dots,x_n)\in\mathbb R^n\) with \(|x_1|\leq Q, |x_i|\leq Q^{-1/(n-1)}\) for \(i=2,\dots,n\). Then \(\det(\Lambda)=1\), \(vol(\mathcal K(Q))=2^n\) so that \(\mathcal K(Q)\) contains a nonzero point. Let us define \(q\) by \(Q=e^{(n-1)q}\). Then \(|x_1|\leq e^{(n-1)q}, |x_i|\leq e^{-q}\) for \(i=2,\dots,n\). Let us consider the convex body \(\mathcal K(q)=T^q(\mathcal K)\) parametrized by \(q\) where \(\mathcal K\) is a given symmetric convex body and \(T: \mathbb R^n\to \mathbb R^n\) is the map with \(T(x_1,x_2,\dots,x_n)=(e^{-(n-1)}x_1, e^{-1} x_2,\dots, e^{-1} x_n)\). Let \(\lambda_i(q)=\lambda_i(\Lambda,\mathcal K(q))=\lambda_i(\Lambda, \mathcal K,q)\) be the Minkowki successive minima of the convex body \(\mathcal K(q)\) seen as functions of \(q\), \(L_i=\log \lambda_i (q)\) and \(\phi_i(q)=L_i(q)/q, \;\overline{\phi_i}= \lim\sup _{q\to\infty} \phi_i(q)\) and \(\underline{\phi_i}= \lim\inf _{q\to\infty} \phi_i(q)\). A lattice \(\Lambda\) will be called \textit{proper} if for every \(j,\;1\leq j\leq n,\) there are arbitrarily large values of \(q\) with \(L_j(q)=L_{j+1}(q)\). Observe that, from [\textit{W. M. Schmidt} and \textit{L. Summerer}, Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)], \(\Lambda( \xi_1,\dots,\xi_{n-1})\) is proper if \(1,\xi_1,\dots,\xi_{n-1}\) are linearly independent over \(\mathbb Q\). The authors prove the following results in parametric Geometry of Numbers: {\parindent=6mm \begin{itemize}\item[-] Suppose that \(n>2\) and \(\Lambda\) is proper. Then \((n-1)\underline{\phi}_1+\overline{\phi}_n\leq -\underline{\phi}_n(\frac{n}{n-2}+\underline{\phi}_1+ \frac{2}{n-2}\overline{\phi}_n)\). Moreover for \(1\leq i <n,\) we have \((n-1)\underline{\phi}_i+\overline{\phi}_n\leq \overline{\phi}_i(n-\underline{\phi}_i+\overline{\phi}_n)\). \item[-] Suppose that \(n>2\) and \(\Lambda\) is proper. Then \((n-1)\overline{\phi}_n+\underline{\phi}_1\geq -\overline{\phi}_1(\frac{n}{n-2}+\overline{\phi}_n+\frac{2}{n-2}\underline{\phi}_1)\). Moreover for \(1\leq j \leq n,\) we have \((n-1)\overline{\phi}_j+\underline{\phi}_1\geq \underline{\phi}_j(n-\overline{\phi}_j+\underline{\phi}_1)\). \item[-] Suppose that \(\Lambda\) is proper. Then we have \[ \underline\phi_1\leq\frac{(n^2-3n+3)\overline\phi_1+(n-3)\overline\phi_1^2}{(n-2)^2+(2n-5)\overline\phi_1+\overline\phi_1^2}\text{ and } \overline\phi_n\geq\frac{(n^2-3n+3)\underline\phi_n+(n-3)\underline\phi_n^2}{(n-2)^2+(2n-5)\underline\phi_n+\underline\phi_n^2}. \] \end{itemize}} Observe that the last majoration of \(\underline \phi_1\) sharpens an inequality of \textit{V. Jarník} [Czech. Math. J. 4(79), 330--353 (1954; Zbl 0057.28303)] in terms of the classical approximation constants \(\omega\) and \(\hat\omega\). The proofs of these inequalities are difficult. They are obtained from the important theorem 4.1 of the article. In particular, the author defines some functions \(P_1, P_2,\dots,P_n\) easier to deal with than \(L_1, L_2,\dots,L_n\) involving the \(i\)th compound \(\Lambda^{(i)}\) of \(\Lambda\) defined from some \textit{exterior products}, some \textit{pseudo-compounds} \(\mathcal K(q)^{(i)}\) of \(\mathcal K(q)\) see [\textit{K. Mahler}, Proc. Lond. Math. Soc., III. Ser. 5, 358--379, 380--384 (1955; Zbl 0065.28002)], the successive minima \(\mu_{i1},\dots, \mu_{i N_i}\) with respect to \(\Lambda^{(i)},\mathcal K(q)^{(i)}\) and \(M_{ik}=\log \mu_{i ,k}\) and the so-called \textit{\((n,\gamma)\)-systems} defined from \(\mathcal P=(P_1,\dots,P_n)\).
scientific article; zbMATH DE number 6131821
scientific article