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On the number of lattice points in convex symmetric bodies and their duals. (English) Zbl 0752.52008
Isr. J. Math. 74, No. 2-3, 347-357 (1991); erratum ibid. 171, 443-444 (2009).
Let \(K\subset\mathbb{R}^ n\) be a convex, centrally symmetric, bounded, absorbing set, \(vol(K)\) its standard volume, \(K^*\) its polar convex set, and \(\# (K\cap\mathbb{Z}^ n)\) the number of lattice points in \(K\). Authors prove that \(\# (K\cap\mathbb{Z}^ n)/(\# (K^*\cap\mathbb{Z}^ n)\text{vol}(K))\) is bounded below and above by positive constants depending on \(n\) but not on \(K\), and they give various applications of this result.

MSC:
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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