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New bounds in some transference theorems in the geometry of numbers. (English) Zbl 0786.11035

Let \(\|\;\|\) and \(d\) be the Euclidean norm and metric on \(\mathbb{R}^ n\), and let \(B_ n\) be the closed unit ball. Let \(L\) be a lattice in \(\mathbb{R}^ n\) (an additive subgroup generated by \(n\) linearly independent vectors) and \(L^*\) the dual lattice: \(L^*=\{u\in\mathbb{R}^ n\): \(uv\in\mathbb{Z}\) for each \(v\in\mathbb{R}^ n\}\), \(uv\) being the Euclidean inner product. We define the covering radius and the successive minima of \(L\) in the usual way: \[ \mu(L)=\min\{r>0:\;L+rB_ n= \mathbb{R}^ n\}, \qquad \lambda_ i(L)=\min\{r>0:\text{ dim span}(L\cap rB_ n)\geq i\} \] for \(i=1,\dots,n\). Let \(\Lambda_ n\) be the family of all lattices in \(\mathbb{R}^ n\). We denote \[ \xi_ n= \sup_{L\in\Lambda_ n} \max_{1\leq i\leq n} \lambda_ i(L) \lambda_{n-i+1}(L^*), \qquad \eta_ n= \sup_{L\in\Lambda_ n} \mu(L) \lambda_ 1(L^*), \]
\[ \zeta_ n= \sup_{L\in\Lambda_ n} \sup_{u\in\mathbb{R}^ n\setminus L} \inf_{\textstyle{{{v\in L^*} \atop {uv\not\in\mathbb{Z}}}}} \| v\| d(u,L)d(uv,\mathbb{Z})^{-1}. \] It is proved that \(\xi_ n\leq n\), \(\eta_ n\leq{1\over 2}n\) and \(\zeta_ n\leq 5n\) for every \(n\); the constant 5 may be replaced by a smaller one. Furthermore, \[ \xi_ n,\eta_ n\leq {n\over {2\pi}} (1+O(n^{-1/2})) \quad\text{as } n\to\infty, \qquad \zeta_ n\leq {{2n}\over \pi}(1+ O(n^{-1/2})) \quad\text{as } n\to\infty. \] As concerns lower bounds, it is known that \[ \xi_ n,\eta_ n\geq {n\over {2\pi e}}(1+o(1)) \quad \text{as }n\to\infty, \qquad \zeta_ n\geq {n\over {\pi e}} \quad \text{as } n\to\infty. \] The proofs of the results obtained are non-constructive; they consist in investigating the properties of the Gaussian-like probability measure \(\sigma_ L(A)= \sum_{x\in A} e^{-\pi x^ 2} / \sum_{x\in L} e^{-\pi x^ 2}\) on a lattice \(L\), and the Fourier transform of \(\sigma_ L\).

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
11H60 Mean value and transfer theorems
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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References:

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