Parallelepipeds in sets of integers
type: Item
instance of: scholarly article
title: Parallelepipeds in sets of integers
zbMATH Open document ID: 0622.10043
DOI: 10.1016/0097-3165(87)90011-2
author: Daniel Berend
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: Given subsets \(A_ 1,A_ 2,...,A_ k\) of integers, let \(\sum^{k}_{i=1}A_ i=\{a_ 1+a_ 2+...+a_ k\); \(a_ i\in A_ i\), \(1\leq i\leq k\}\). A set P of positive integers is called a d- dimensional parallelepiped if it is of the form \(P=m+\sum^{d}_{i=1}\{0,m_ i\}\) for some positive integers \(m,m_ 1,...,m_ d\). If moreover \(m_ 1| m_ 2|...| m_ d\) the parallelepiped is called arithmetical. For any positive integers d and N, let \(P_ d(N)\) be the maximal cardinality of a subset of \(\{\) 1,2,...,N\(\}\) not containing a d-dimensional parallelepiped, and let \(A_ d(N)\) be the corresponding number for arithmetic parallelepipeds. The author proves the following upper bounds \[ P_ d(N)\quad <\quad N^{1-1/2^{d-1}}\quad +\quad O(N^{3/4-1/2^{d-1}})\quad, \] \[ A_ d(N)\quad <\quad \beta_ d N^{1-1/(2^ d-1)}\quad +\quad O(N^{1- 2/(2^ d-1)}) \] with explicitly given \(\beta_ d\).
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 4008511
zbMATH Keywords: Sidon set
zbMATH Keywords: maximal cardinality
zbMATH Keywords: arithmetic parallelepipeds
reviewed by: Štefan Porubský
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90011-2
OpenAlex ID: W2032728802
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description: scientific article; zbMATH DE number 4008511