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Parametric geometry of numbers and applications. (English) Zbl 1236.11060

This article deals with classical geometry of numbers. Let \(\mu_1,\dots,\mu_n\) be reals with \(\mu_1+\dots+\mu_n=0\). For \(Q>1\) let \(T_Q:\mathbb R^n\to\mathbb R^n\) be the linear map with \[ \mathbf p:=(p_1,\dots, p_n)\to (Q^{\mu_1}p_1,\dots, Q^{\mu_n} p_n). \] A symmetric convex body \(K\) gives rise to the bodies \(K(Q)=T_Q(K)\) parametrized by \(Q\).
The authors study the successive minima \(\lambda_1(Q),\dots,\lambda_n(Q)\) with respect to \(K(Q)\), \(\Lambda\) seen as a function of \(Q\). Each \(\lambda_i(Q)\) is a continuous function of \(Q\) since \(Q\) is closed. They wonder whether for given \(s, 1\leq s\leq n\), there are arbitrarily large values of \(Q\) with \(\lambda_s(Q)=\lambda_{s+1}(Q)\). When \(A=\{i_1<\dots<i_s\}\subset \{1,\dots,n\}\), set \(\mu_A=\sum_{i\in A}\mu_i\) and let \(\pi_A: \mathbb R^n\to \mathbb R^s\) be the map with \(\pi_A(\mathbf p)=(p_{i_1},\dots,p_{i_s})\in\mathbb R^s\).
The authors prove: Suppose that for every \(s\)-dimensional space \(S\) spanned by lattice points (i.e. points of \(\Lambda\)), there is some \(A\) of cardinality \(s\) with \(\mu_A<0\) and \(\pi_A(S)=\mathbb R^s\). Then there are arbitrary large values of \(Q\) with \(\lambda_s(Q)=\lambda_{s+1}(Q)\).
In general there is a nonzero lattice point \(\mathbf p\) in \(\lambda_1(Q)K(Q)\). Next the authors define \(\psi_i(Q)\) for \(Q>1\) by \[ \lambda_i(Q)=Q^{\psi_i(Q)}\text{\;for\;} i=1,\dots,n. \] The \(\psi_i(Q)\) are again continuous, one has \(0<\psi_1(Q)\leq \dots\leq \psi_n(Q)\) and from Minkowski’s theorem \[ |\psi_1(Q)+\dots+\psi_n(Q)|\leq c(K,\Lambda)/\log Q, \] for a certain constant depending on \(\Lambda\) and \(K\). The authors show that the quantities \[ \overline{\psi_i}=\lim \sup_{Q\to\infty}\psi_i(Q) \text{ and } \underline{\psi_i}=\lim \inf_{Q\to\infty}\psi_i(Q) \] are finite and satisfy the inequalities \(\overline{\psi_1}\leq \dots\leq \overline{\psi_n}\) and \(\underline{\psi_1}\leq \dots\leq \underline{\psi_n}\) and also \(\overline{\psi_i}\geq \underline{\psi_i},\;i=1,\dots, n\).
They prove the following theorem: For \(1\leq i\leq n\) one has \[ \begin{aligned} &\overline{\psi}_1+\dots+\overline{\psi}_{i-1}+\underline{\psi}_i+\overline{\psi}_{i+1}+\dots+\overline{\psi}_n\geq 0,\\ &\underline{\psi}_1+\dots+\underline{\psi}_{i-1}+\overline{\psi}_i+\underline{\psi}_{i+1}+\dots+\underline{\psi}_n\leq 0. \end{aligned} \] They also give some application for the case \(n=3\) corresponding to the dimension two in V. Jarník [ Trav. Inst. Math. Tbilissi 3, 193–212 (1938; Zbl 0019.10602)] and M. Laurent [Can. J. Math. 61, No. 1, 165–189 (2009; Zbl 1229.11101)] papers in the context of Dirichlet simultaneous approximation. When \(n=3\), then \[ \begin{aligned} &\overline{\psi}_1+\underline{\psi}_3+2\overline{\psi}_1\underline{\psi}_3=0,\\ &2\underline{\psi}_1+\overline{\psi}_3\leq -\underline{\psi}_3(3+2\underline{\psi}_1+4\overline{\psi}_3),\\ &2\overline{\psi}_3+\underline{\psi}_1\geq -\overline{\psi}_1(3+2\overline{\psi}_3+4\underline{\psi}_1).\end{aligned} \]

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
11J13 Simultaneous homogeneous approximation, linear forms
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