## 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

### Keywords:

lattices; successive minima; simultaneous approximation

### Citations:

Zbl 0019.10602; Zbl 1229.11101
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