On the dimension of the Hilbert cubes. (English) Zbl 0989.11012

Given \(k\geq 1\), a set \(H\subset \mathbb{N}\) is called a cube of size \(k\) if there exist \(a>0\) and \(x_1,\dots,x_k\) such that \(H=\left\{a+\sum_{i=1}^k\varepsilon_ix_i:\varepsilon=0\text{ or }1\right\}\). The author proves that (1) there is an infinite sequence \(A\subset\mathbb{N}\) of positive lower density and containing in \(A\cap\{1,\dots,n\}\) the largest cube of size \(\leq c\sqrt{\log n\log\log n}\), where \(c=4\cdot(\log(4/3))^{-1/2}\); (2) if \(A\) is a random sequence in \(\mathbb{N}\) with \(\text{Pr}(a\in A)=p\), then with probability \(1\) the largest size of a cube here is \(>c_p\sqrt{\log n}\).


11B75 Other combinatorial number theory
11B05 Density, gaps, topology
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