## 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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