New examples of noncommutative \(\Lambda(p)\) sets
Since 1960 and \textit{W. Rudin}'s paper [J. Math. Mech. 9, 203--227 (1960; Zbl 0091.05802)], the notion of \(\Lambda(p)\) set has generated numerous works; for example: \textit{G. Pisier} [Sémin. Géom. des Espaces de Banach 1977--1978, Exposés No. 12--13 (1978; Zbl 0388.43009)], \textit{J. Bourgain} [Acta Math. 162, 227--245 (1989; Zbl 0674.43004)], see also \textit{M. Talagrand} [Acta Math. 175, 273--300 (1995; Zbl 0917.46006)]. For a survey of interactions of this notion with various parts of mathematics, see \textit{J. Bourgain} [\(\Lambda_p\)-sets in analysis: Results, problems and related aspects, Handbook of the geometry of Banach spaces, Volume 1, Amsterdam: Elsevier, 195--232 (2001; Zbl 1016.43004)]. For \(2<p<+\infty\), a subset \(E\subseteq {\mathbb Z}\) is called a \(\Lambda(p)\) set if the \(L^p\) norm is equivalent to the \(L^2\) norm on \(L^p_E({\mathbb T})\) (where \({\mathbb T}={\mathbb R}/2\pi{\mathbb Z}\) and, as usual \(L^p_E({\mathbb T})= \{f\in L^p({\mathbb T})\); \(\widehat f(n)=0\) for \(n\in {\mathbb Z}\setminus E\}\)). In other words, there exists a constant \(C>0\) such that \(\| f\|_{L^p}\leq C(\sum_{n\in E} | \widehat f(n)|^2)^{1/2}\) for every \(f\in L^p_E({\mathbb T})\). In her thesis [\textit{A. Harcharras}, Stud. Math. 137, 203--260 (1999; Zbl 0948.43002)], the second author, introduced the stronger notion of noncommutative \(\Lambda(p)\) set (or \(\Lambda(p)_{cb}\) set): let \(S_p\) be the Schatten \(p\)-class over the Hilbert space \(\ell_2\); a subset \(E\subseteq {\mathbb Z}\) is a noncommutative \(\Lambda(p)\) set if there exists a constant \(C>0\) such that \[ \| f\|_{L^p(S_p)}\leq C\,\max\Biggl\{\biggl\| \biggl(\sum_{n\in E} \widehat f(n)^\ast\,\widehat f(n)\biggr)^{1/2} \biggr\|_{S_p}, \biggl\| \biggl(\sum_{n\in E} \widehat f(n)\,\widehat f(n)^\ast\biggr)^{1/2} \biggr\|_{S_p}\Biggr\}\,, \] for every \(f\in L^p_E(S_p)\). For the study of \(\Lambda(p)\) sets, Rudin used a combinatorial property, called now \(Z^+(k)\) (where \(k\geq 1\) is a fixed integer); it says that the number of \(k\)-tuples \((n_1,\ldots,n_k)\in E^k\) with which we can write \(N=n_1+\cdots+n_k\) (\(N\in {\mathbb Z}\)), is uniformly bounded with respect to \(N\). Every set with this property is a \(\Lambda(2k)\) set. In her thesis, A. Harcharras showed that this property does not suffice to ensure that \(E\) is a noncommutative \(\Lambda(2k)\) set, and introduced the property called \(Z(k)\): \(E\) has this property if the number of \(k\)-tuples \((n_1,\ldots,n_k)\in E^k\) with \(n_1,\ldots, n_k\) all \textit{distinct}, with which we can write \(N=\sum_{j=1}^k (-1)^{j+1} n_j\) (\(N\in {\mathbb Z}\)), is uniformly bounded with respect to \(N\). She showed that \(E\) is \(\Lambda(2k)_{cb}\) when \(E\) has property \(Z(k)\). In the paper under review, the authors consider, for a given finite set \(Q\) of prime numbers, the set \(E_Q\) of all the positive integers whose every prime factor is in \(Q\) (i.e. the set which is multiplicatively generated by \(Q\)). When \(Q\) consists only of one prime, \(E_Q\) is a Hadamard set, and so a Sidon set; but when \(Q\) has at least two elements, \(E_Q\) is never a Sidon set. Moreover, the authors give a simple example (\(Q=\{2,3\}\)) such that \(E_Q\) does not have property \(Z(3)\). The authors introduce a new property, called \(Z^\star(k)\), which is a bit complicated to be reproduced here, and their main result is that every set \(E\subseteq {\mathbb Z}\) with property \(Z^\star(k)\) is a \(\Lambda(2k)_{cb}\) set (Theorem 4). Moreover, it follows from a deep result of \textit{H. P. Schlickewei} [J. Reine Angew. Math. 406, 109--120 (1990; Zbl 0693.10016)], that \(E_Q\) has both properties \(Z^+(k)\) and \(Z^\star(k)\) for all \(k\geq 2\); consequently, \(E_Q\) is a \(\Lambda(p)_{cb}\) set for every real number \(p>2\). The paper is very carefully written, with a detailed introduction part, so that it can be read by non-specialists.
scientific article; zbMATH DE number 2028859