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The number of lattice points in a spherical shell. (English. Russian original) Zbl 1126.11322

Proc. Steklov Inst. Math. 239, 315-319 (2002); translation from Tr. Mat. Inst. Im. V. A. Steklova 239, 332-335 (2002).
From the text: New dependences between lattices and their duals are established. In Euclidean spaces of large dimensions, an exponential lower bound for the number of points of a lattice \(L\) that lie in a spherical layer with close inner and outer radii is obtained. The radii are reciprocal to the packing radius of the dual lattice \(L'\).
Here the author proves the following:
If \(R_1 = {{q_1}\over {2\pi r(L^\prime)}},\;R_2 = {{q_2}\over {2\pi r(L^\prime)}}\), then the \(L\)-translates of the annulus cover \(\mathbb{R}^n\), \[ \bigcup_{\vec{\mu}\in L} S_{R_1,\,R_2}(\vec{\mu}) \supset \mathbb{R}^n, \] and the number \(N_{R_1, R_2} = \sum_{\vec{\mu}\in L; R_1 \leq |\vec{\mu} | \leq R_2} 1\) of lattice points in the \(n\)-dimensional annulus is estimated from below (with the same \(R_i\) as before) by \[ N_{R_1,\,R_2} \geq 2, \text{ and } N_{R_1, R_2} \gg \left({1\over2} e \right)^n \text{ for } n\to\infty. \]
For the entire collection see [Zbl 1059.52002].

MSC:

11H16 Nonconvex bodies
11H31 Lattice packing and covering (number-theoretic aspects)
11P21 Lattice points in specified regions
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