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

Hegyvári, Norbert

J. Number Theory 77, No. 2, 326-330 (1999).

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 \(>c_p\sqrt{\log n}\).

Reviewer: Štefan Porubský (Praha)

MSC:

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

Keywords:

Ramsey-type theorems; Hilbert cube; combinatorial cube

References:

[1]	Hilbert, D., Über die irreducibilität rationaler functionen mit ganzzäligen koefficienten, J. reine angew. math., 110, 104-109, (1892) · JFM 24.0087.03
[2]	P. Erdös, A. Sárközy, and V. T. Sós, On a conjecture of Roth and related problems I, Algorithms Combin.8, 47-59.
[3]	Graham, R.L.; Rothschild, B.; Spencer, J., Ramsey theory, (1980), Wiley Interscience
[4]	Hegyvári, H., On representation problems in the additive number theory, Acta math. hung., 72, 35-44, (1996) · Zbl 0866.11015
[5]	N. Hindman, Ultrafilters and combinatorial number theory, in Number Theory Carbonadle (M. Nathanson, Ed.), Lecture Notes in Math., Vol. 751, pp. 119-184. · Zbl 0416.10042
[6]	N. Hegyvári, and, A. Sárközy, On the size of a combinatorial cube of certain sets, to appear.
[7]	Erdös, P.; Rényi, A., On a new law of large numbers, J. anal. math., 22, 103-111, (1970) · Zbl 0225.60015