Combinatorial dimension and random sets
It is well-known that the ordinary n-fold Cartesian product of a dissociate subset F of a discrete Abelian group \(\Gamma\) is p-Sidon precisely for \(2\geq p\geq 2n/(n+1)\) (compare e.g. [\textit{G. W. Johnson}, \textit{G. S. Woodward}, Indiana Univ. Math. J. 24, 161-167 (1974; Zbl 0285.43006)]. Let us describe this shortly by saying that \(F^ n\) is exactly \(2n/(n+1)-Sidon.\) To produce sets which are exactly p-Sidon for p in the interval (2(n- 1)/n, \(2n/(n+1)\), the first author had constructed ''fractional'' Cartesian products of dissociate sets. This has motivated the following notion of combinatorial dimension. If E is a subset of \({\mathbb{N}}^ J\), and if \(s\in {\mathbb{N}}\), then let \(\psi_ E(s)=\max \{| E\cap(A_ 1x...xA_ J)|:A_ i\subset {\mathbb{N}},| A_ 1| =...=| A_ J| =s\}.\) The combinatorial dimension of E is given by \(\dim E=\inf \{a:\overline{\lim}_{s\to \infty}\frac{\psi_ E(s)}{s^ a}<\infty \}.\) Moreover, the constructions of the first author had lead to the following question: Given \(1<\alpha<2\), are there \(\alpha\)-dimensional sets in \({\mathbb{N}}^ 2?\) This question is affirmatively answered in this paper by even stronger results.
scientific article; zbMATH DE number 3846186