Intermediate Diophantine exponents and parametric geometry of numbers (Q2889256)

Let NEWLINE\[NEWLINE \Theta:= \left(\begin{matrix} \theta_{11}& \dots& \theta_{1m}\\ \vdots& \vdots& \vdots\\ \theta_{n1}& \dots& \theta_{nm} \end{matrix},\right) \;\;\theta_{ij}\in\mathbb R, n+m\geq 3,NEWLINE\]NEWLINE be a matrix. Consider the system of linear equations NEWLINE\[NEWLINE\Theta \mathbf x=\mathbf y\tag{1}NEWLINE\]NEWLINE with variables \(\mathbf x\in \mathbb R^m\) and \(\mathbf y\in \mathbb R^n.\) Let \(|\cdot|\) the sup norm in the corresponding space. The supremum of the real numbers \(\gamma\) such that there are arbitrarily large values of \(t\) for which (resp., such that for every \(t\) large enough) the system of inequalities NEWLINE\[NEWLINE|\mathbf x|\leq t, |\Theta\mathbf x-\mathbf y|\leq t^{-\gamma}\tag{2}NEWLINE\]NEWLINE has a nonzero solution in \((\mathbf x,\mathbf y) \in\mathbb Z^m\oplus\mathbb Z^n\) is called the \textit{regular} (resp. \textit{uniform} ) \textit{Diophantine exponent} of \(\Theta\).NEWLINENEWLINEThis paper is a generalization to the case of approximating the space of solutions to ({1}) by \(p\)-dimensional rational subspaces of \(\mathbb R^{m+n}\) as done by \textit{W. M. Schmidt} [Ann. Math. (2) 85, 430--472 (1967; Zbl 0152.03602)]. Later \textit{M. Laurent} [in: Analytic number theory. Essays in honor of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press, 306--314 (2009; Zbl 1163.11053)], and with \textit{Y. Bugeaud} [Math. Z. 265, No. 2, 249--262 (2010; Zbl 1234.11086)], gave a corresponding definition in the case when \(m = 1\).NEWLINENEWLINEThe supremum of the real numbers \(t\) such that there are arbitrarily large values of \(t\) for which (resp., such that for every \(t\) large enough) the system of inequalities \(|\mathbf x|\leq t, |\Theta\mathbf x-\mathbf y|\leq t^{-\gamma}\) has \(p\) solutions \(\mathbf z_i = (\mathbf x_i,\mathbf y_i) \in \mathbb Z^m \oplus \mathbb Z^n, i = 1,\dots,p\), linearly independent over \(\mathbb Z\), is called the \textit{pth regular} (resp. \textit{uniform}) \textit{Diophantine exponent of the first type} of \(\Theta\).NEWLINENEWLINEThe author proposes a definition of intermediate exponents of the second type, which is consistent with Laurent's and formulates his main results on these quantities. A part of the article is devoted to the exponents naturally emerging in the \textit{parametric geometry of numbers} developed by \textit{W. M. Schmidt} and \textit{L. Summerer} [Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)]. Those exponents are closely connected to the Diophantine exponents. It allows reformulating the main results in terms of Schmidt-Summerer's exponents, a part we choose to detail here:NEWLINENEWLINELet \(\Lambda\) be a unimodular \(d\)-dimensional lattice in \(\mathbb R^d\). Denote by \(\mathcal B_\infty^d\) the unit ball in sup-norm, i.e., the cube with vertices at the points \((\pm 1,\dots,\pm 1)\). For each \(d\)-tuple \(\tau=(\tau_1,\dots,\tau_d)\in \mathbb R^d\), denote by \(D_\tau\) the diagonal \(d\times d\) matrix with \(e^{\tau_1},\dots,e^{\tau_d}\) on the main diagonal. Let \(\lambda_p(M)\) denote the \(p\)th successive minimum of a compact symmetric convex body \(M \subset \mathbb R^d\) (centered at the origin) with respect to the lattice \(\Lambda\). Suppose we have a path \(\mathcal T\) in \(\mathbb R^d\) defined as \(\tau=\tau(s), s\in\mathbb R_+\) such that \(\tau_1(s)+\dots+\tau_d(s)=0\) for all \(s\). Set \(\mathcal B(s)=D_{\tau(s)}\mathcal B_\infty^d\). For each \(p=1,\dots,d\) consider the two functions \(\psi_p(\Lambda,\mathcal T,s):=\frac{\ln(\lambda_p(\mathcal B(s)))}{s}\), \(\Psi_p(\Lambda, \mathcal T ,s):=\sum_{i=1}^p \psi_i(\Lambda,\mathcal T,s).\)NEWLINENEWLINEThe quantities \({\underline{\psi}}_p(\Lambda,\mathcal T)=\lim \inf_{s\rightarrow\infty} \psi_p(\Lambda,\mathcal T,s)\) and \(\overline{\psi}_p(\Lambda,\mathcal T)=\lim \sup_{s\rightarrow\infty} \Psi_p(\Lambda,\mathcal T,s)\) are called the \(p\)th \textit{lower and upper Schmidt-Summerer exponents of the first type} and the quantities \(\underline{\Psi}_p(\Lambda,\mathcal T)=\lim \inf_{s\rightarrow\infty} \psi_p(\Lambda,\mathcal T,s)\) and \(\overline{\Psi}_p(\Lambda,\mathcal T)=\lim \sup_{s\rightarrow\infty} \Psi_p(\Lambda,\mathcal T,s)\) are called the \(p\)th \textit{lower and upper Schmidt-Summerer exponents of the second type}. If the context is clear we note \(\psi_p(s), \Psi_p(s), \underline{\psi}_p,\overline{\psi}_p,\underline{\Psi}_p,\overline{\Psi}_p\). The author proves:NEWLINENEWLINE Theorem. Suppose that the space of integer solutions of (1) is not a one-dimensional lattice. ThenNEWLINENEWLINE\(\underline{\Psi}_2\leq 2\underline{\Psi}_1+d\frac{\overline{\Psi}_1-\underline{\Psi}_1}{n+n\overline{\Psi}_1}\) if \(\overline{\Psi}_1\not=-1\) andNEWLINENEWLINE\(\underline{\Psi}_2\leq 2\underline{\Psi}_1+d\frac{\overline{\Psi}_1-\underline{\Psi}_1}{m-n\overline{\Psi}_1}\).NEWLINENEWLINE{Theorem. } We have \(\overline{\Psi}_2\leq \frac{(d-2)\overline{\Psi}_1}{(n-1)+n\overline{\Psi}_1}\) if \(\overline{\Psi}_1\geq \frac{m-n}{2n}\) and \(\overline{\Psi}_2\leq \frac{(d-2)\overline{\Psi}_1}{(m-1)-n\overline{\Psi}_1}\) if \(\overline{\Psi}_1\leq \frac{m-n}{2n}\).