17 research outputs found

    A simple proof of a reverse Minkowski theorem for integral lattices

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    We prove that for any integral lattice L⊂Rn\mathcal{L} \subset \mathbb{R}^n (that is, a lattice L\mathcal{L} such that the inner product ⟨y1,y2⟩\langle \mathbf{y}_1,\mathbf{y}_2 \rangle is an integer for all y1,y2∈L\mathbf{y}_1, \mathbf{y}_2 \in \mathcal{L}) and any positive integer kk, ∣{y∈L : ∥y∥2=k}∣≤2(n+2k−22k−1)  , |\{ \mathbf{y} \in \mathcal{L} \ : \ \|\mathbf{y}\|^2 = k\}| \leq 2 \binom{n+2k-2}{2k-1} \; , giving a nearly tight reverse Minkowski theorem for integral lattices

    Kissing numbers and transference theorems from generalized tail bounds

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    We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. From this we obtain quite general transference bounds, as well as bounds on the number of lattice points contained in certain bodies. As applications, we bound the lattice kissing number in ℓp\ell_p norms by e(n+o(n))/pe^{(n+ o(n))/p} for 0<p≤20 < p \leq 2, and also give a proof of a new transference bound in the ℓ1\ell_1 norm.Comment: Previous title: "Generalizations of Banaszczyk's transference theorems and tail bound

    Universal quadratic forms and Dedekind zeta functions

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    We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field KK, under the assumption that the codifferent of KK is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields.Comment: 12 pages. Preprin

    On lattice extensions

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    A lattice Λ\Lambda is said to be an extension of a sublattice LL of smaller rank if LL is equal to the intersection of Λ\Lambda with the subspace spanned by LL. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.Comment: 18 pages, 4 figures; to appear in Monatshefte f\"ur Mathemati
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