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On the Dimension of the Hilbert Cubes

✍ Scribed by Norbert Hegyvári

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
90 KB
Volume
77
Category
Article
ISSN
0022-314X

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✦ Synopsis

A sequence of positive integers with positive lower density contains a Hilbert (or combinatorial) cube size c log log n up to n. We prove that c log log n cannot be replaced by c$ -log n log log n.

1999 Academic Press

In [1] D. Hilbert showed (using different terminology) that for any k 1, if N is finitely colored than there exists in one color infinitely many translates of a k-cube. We call H/N a k-cube if there exist a>0 and x 1 , x 2 , ..., x k such that H=H(a, x 1 , ..., x k )= { a+ : k i=1 = i a i : = i =0 or 1 = .

This result was essentially the first Ramsey-type theorem. The density version of the Hilbert result is the following: Theorem A: Let k 1 be a integer and assume A/[1, n] and |A| n 1&2 &k . Then A contains a k-cube. (see [2], [3]). This result implies the following Corollary. Let A be an infinite sequence of integers with d(A)= lim n Ä inf A(n)Ân>0,

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