## 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 $$>c_p\sqrt{\log n}$$.

### MSC:

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

### Keywords:

Ramsey-type theorems; Hilbert cube; combinatorial cube
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### References:

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