On lattice coverings by simplices

Xue, Fei; Zong, Chuanming

doi:10.1515/advgeom-2017-0049

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…where c is some absolute constant (for the upper bound see [15, p. 19]; the lower bound was recently derived in [21]). The definitions for coverings of Z n with a finite set K ⊂ Z n are similar, with vol(K) replaced by |K|.…”

Section: Additional Remarks and Open Problemsmentioning

confidence: 99%

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Improved Bounds on Sidon Sets via Lattice Packings of Simplices

Kovačević¹,

Tan²

2017

SIAM J. Discrete Math.

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Abstract.A B h set (or Sidon set of order h) in an Abelian group G is any subset {b0, b1, . . . , bn} of G with the property that all the sums bi 1 + · · · + bi h are different up to the order of the summands. Let φ(h, n) denote the order of the smallest Abelian group containing a B h set of cardinality n + 1. It is shown thatwhere δl(△ n ) is the lattice packing density of an n-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known (n ≤ 3) and gives improved bounds on φ(h, n) in the remaining cases. The corresponding geometric characterization of bases of order h in finite Abelian groups in terms of lattice coverings by simplices is also given.

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Section: Additional Remarks and Open Problemsmentioning

confidence: 99%

“…The authors would like to thank the anonymous referees for a thorough review and many helpful comments on the original version of the manuscript, as well as for pointing out the work [21].…”

Section: Acknowledgmentmentioning

confidence: 99%

Improved Bounds on Sidon Sets via Lattice Packings of Simplices

Kovačević¹,

Tan²

2017

SIAM J. Discrete Math.

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“…where the upper bound was discovered by C. M. Fiduccia, R. W. Forcade and J. S. Zito [9] and R. Dougherty and V. Faber [4] in the 1990s by constructing a particular lattice covering and the lower bound was achieved by F. Xue and C. Zong [22] in 2018. In fact, it was proved by R. Forcade and J. Lamoreaux [10] and R. Dougherty and V. Faber [4] that the upper bound is a local minimum.…”

Section: Introductionmentioning

confidence: 96%

“…Since the 1960s, progress in covering is very limited (see [2]). In 2018, F. Xue and C. Zong [22] discovered that…”

Section: Introductionmentioning

confidence: 99%