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 \(\sigma_ L\).


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)
Full Text: DOI EuDML


[1] Babai, L.: On Lov?sz’ lattice reduction and the nearest lattice point problem. Combinatorica6, 1-13 (1986) · Zbl 0593.68030
[2] Banaszczyk, W.: Closed subgroups of nuclear spaces are weakly closed. Studia Math.80, 119-128 (1984) · Zbl 0568.46002
[3] Banaszczyk, W.: Pontryagin duality for subgroups and quotients of nuclear spaces. Math. Ann.273, 653-664 (1986) · Zbl 0593.46006
[4] Banaszczyk, W.: Polar lattices from the point of view of nuclear spaces. Rev. Mat. Univ. Complutense Madr.2 (special issue), 35-46 (1989) · Zbl 0745.46005
[5] Banaszczyk, W.: Additive subgroups of topological vector spaces. (Lect. Notes Math., vol 1466) Berlin Heidelberg New York: Springer 1991 · Zbl 0743.46002
[6] Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin G?ttingen Heidelberg: Springer 1959 · Zbl 0086.26203
[7] Hastad, J.: Dual vectors and lower bounds for the nearest lattice point problem. Combinatorica8, 75-81 (1988) · Zbl 0653.10026
[8] Hewitt, E., Ross, K.A.: Abstract harmonic analysis, vol. II. Berlin Heidelberg New York: Springer 1970 · Zbl 0213.40103
[9] Khinchin, A.I.: A quantitative formulation of Kronecker’s theory of approximation. Izv. Akad. Nauk SSSR, Ser. Mat.12, 113-122 (1948) · Zbl 0030.02002
[10] Lagarias, J.C., Lenstra, H.W., Schnorr, C.P.: Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica10, 333-348 (1990) · Zbl 0723.11029
[11] Mahler, K.: Ein ?bertragungsprinzip f?r konvexe K?rper. ?as. P?stov?n? Mat. Fys.68, 93-102 (1939)
[12] Milnor, J., Husemoller, D.: Symmetric bilinear forms. Berlin Heidelberg New York: Springer 1973 · Zbl 0292.10016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.