Parallelepipeds in sets of integers

Daniel Berend

doi:10.1016/0097-3165(87)90011-2

Parallelepipeds in sets of integers

Daniel Berend

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

A d-dimensional parallelepiped in N is a set of the form {m + ∑_{i ε{lunate} S}m_i: S ⊆ {1, 2,..., d}} for some positive integers m, m₁, m₂,..., m_d. It is proved that a subset of {1, 2,..., N} not containing a d-dimensional parallelepiped is of cardinality not exceeding N^{1 - 1 2^{d - 1}} + O(N^{3 4 - 1 2^{d - 1}}). A result of a similar nature is established for parallelepipeds satisfying m₁|m₂| ... | m_d.

Original language	English
Pages (from-to)	163-170
Number of pages	8
Journal	Journal of Combinatorial Theory - Series A
Volume	45
Issue number	2
DOIs	https://doi.org/10.1016/0097-3165(87)90011-2
State	Published - 1 Jan 1987

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

10.1016/0097-3165(87)90011-2

Cite this

@article{89a148f0150f4d6eb52da3de8993fb08,

title = "Parallelepipeds in sets of integers",

abstract = "A d-dimensional parallelepiped in N is a set of the form {m + ∑i ε{lunate} Smi: S ⊆ {1, 2,..., d}} for some positive integers m, m1, m2,..., md. It is proved that a subset of {1, 2,..., N} not containing a d-dimensional parallelepiped is of cardinality not exceeding N1 - 1 2d - 1 + O(N 3 4 - 1 2d - 1). A result of a similar nature is established for parallelepipeds satisfying m1|m2| ... | md.",

author = "Daniel Berend",

year = "1987",

month = jan,

day = "1",

doi = "10.1016/0097-3165(87)90011-2",

language = "English",

volume = "45",

pages = "163--170",

journal = "Journal of Combinatorial Theory - Series A",

issn = "0097-3165",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - Parallelepipeds in sets of integers

AU - Berend, Daniel

PY - 1987/1/1

Y1 - 1987/1/1

N2 - A d-dimensional parallelepiped in N is a set of the form {m + ∑i ε{lunate} Smi: S ⊆ {1, 2,..., d}} for some positive integers m, m1, m2,..., md. It is proved that a subset of {1, 2,..., N} not containing a d-dimensional parallelepiped is of cardinality not exceeding N1 - 1 2d - 1 + O(N 3 4 - 1 2d - 1). A result of a similar nature is established for parallelepipeds satisfying m1|m2| ... | md.

AB - A d-dimensional parallelepiped in N is a set of the form {m + ∑i ε{lunate} Smi: S ⊆ {1, 2,..., d}} for some positive integers m, m1, m2,..., md. It is proved that a subset of {1, 2,..., N} not containing a d-dimensional parallelepiped is of cardinality not exceeding N1 - 1 2d - 1 + O(N 3 4 - 1 2d - 1). A result of a similar nature is established for parallelepipeds satisfying m1|m2| ... | md.

UR - http://www.scopus.com/inward/record.url?scp=50849149183&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(87)90011-2

DO - 10.1016/0097-3165(87)90011-2

M3 - Article

AN - SCOPUS:50849149183

SN - 0097-3165

VL - 45

SP - 163

EP - 170

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 2

ER -

Parallelepipeds in sets of integers

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this