Parallelepipeds in sets of integers
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 \(\beta_ d\).
scientific article; zbMATH DE number 4008511