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.


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