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Lattices with few distances. (English) Zbl 0737.11017

The authors consider the Erdős number \(E\) of a lattice \(\Lambda\) in \(n\)-dimensional Euclidean space, \(n\geq 3\), which is defined as follows. For \(x>0\) let \(P(x)\) be the number of values taken in \([0,x]\) by the quadratic form corresponding to \(\Lambda\), and set \(E=F d^{1/n}\) where \(d=\hbox{det} \Lambda\) and \(F\) = limit of \(P(x)/x\) as \(x\to\infty\). Let \(n\) be fixed. The main result of the paper says that \(E\) becomes minimal (and so \(\Lambda\) has “fewest distances”) if and only if \(\Lambda\) is, up to a scalar factor, an even lattice with minimal \(d(=1,2,3\) or 4 depending on \(n\mod 8)\). The key to the proof lies in the observation that, when \(\Lambda\) is integral, \(E\) depends only on which numbers are represented by the genus of \(\Lambda\). For \(n=3\) this is based on a difficult result about the exceptions (to the local-global principle) for which the authors refer to W. Duke and R. Schulze-Pillot [Invent. Math. 99, No.1, 49-57 (1990; Zbl 0692.10020)] and, in particular, to an argument quoted from a letter of Schulze-Pillot. The paper then proceeds with the local computations. There are also tables of lattices with relatively small \(E\) for \(n=3\) and 4.

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52C10 Erdős problems and related topics of discrete geometry

Citations:

Zbl 0692.10020
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Full Text: DOI

Online Encyclopedia of Integer Sequences:

Maximal size of an n-distance set in the plane.

References:

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