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On some random thin sets of integers. (English) Zbl 1131.43007

The paper under review is a complement to [J. Anal. Math. 86, 105–138 (2002; Zbl 1018.43004)], due to the same three authors, which is a remarkable paper in the theory of thin sets in harmonic analysis (to know more on the subject until the mid-eighties, see the survey due to Dechamps-Gondim [M. Dechamps, Publ. Math. Orsay 84-01, Exp. No.7, 51 p. (1984; Zbl 0537.43018)] and you can find some more recent developments in the monograph of [D. Li and H. Queffélec, Introduction to the study of Banach spaces. Analysis and probability. (Introduction à l’étude des espaces de Banach. Analyse et probabilités.) Cours Spécialisés (Paris) 12. Paris: Société Mathématique de France. (2004; Zbl 1078.46001)]). This theory has long suffered from a lack of examples: the known examples were more or less sums of Hadamard sets, so the Banach properties of \(C_\Lambda=\{f\in C| \, \forall n\notin\Lambda,\, \hat f(n)=0\}\) were very rigid; for instance, in this case, \(\Lambda\) is a Rosenthal set hence \(c_0\not\subset C_\Lambda\), i.e. \(C_\Lambda\) does not contain an isomorphic copy of \(c_0\). The first named author was the first to construct, via probabilistic methods, a set of different nature, namely a set \(E\subset\mathbb N\), which is a \(\Lambda(p)\) set for every \(p\) (thin in this sense) and \(c_0\subset C_E\) (so big at the time). Exploiting this idea in a much deeper way, the three authors built in their paper quoted above [Zbl 1018.43004] some sets \(\Lambda\) which belong to several classes of thin sets but such that \(\Lambda\) is uniformly distributed and \(c_0\subset C_\Lambda\). The construction relies on the so-called method of selectors : it consists in choosing some sets \(\Lambda(\omega)=\{n\in\mathbb N| \, \epsilon_n(\omega)=1\}\), where the \(\epsilon_n\) are i.i.d. random variables, with values in \(\{0,1\}\), with expectation \(\delta_n\) (see D. Li and H. Queffélec, op. cit., to know more). The subtility of the behavior of \(\Lambda(\omega)\) depends on the choice of the \(\delta_n\): the smaller the \(\delta_n\) are, the thinner are the \(\Lambda(\omega)\) (almost surely). In the paper under review, there are two parts. In the first one, the authors reprove in a simpler way a theorem contained in their first paper. The proof relies on a new deviation inequality (Theorem 2.1) which is a nice (and useful) application of a recent result due to S. Boucheron, G. Lugosi and P. Massart [Ann. Probab. 31, No. 3, 1583–1614 (2003; Zbl 1051.60020)], and is of independent interest. More precisely, Theorem 3.1. (Theorem II.7 in their first paper) states: There exists a subset \(\Lambda\subset\mathbb N\) which is uniformly distributed and contains a subset \(E\) with the following properties: 1) \(E\) is a \(UC\) set, a \(\Lambda(q)\) set for every \(q\) and a \(4/3\)-Rider set, but is not \(q\)-Rider for \(q<4/3\). 2) \(E\) has a positive upper density inside \(\Lambda\); in particular \(c_0\subset C_E\) and \(E\) is not a Rosenthal set. In the second part, they answer negatively a question left open in their first paper. Theorem 4.1 states: Let \(p\in]4/2,2[\) and \(\alpha=\frac{2(p-1)}{2-p}>1\). The random set \(\Lambda=\Lambda(\omega)\) corresponding to the selectors of mean \(\delta_n=c\frac{(\log k)^\alpha}{k(\log\log k)^{\alpha+1}}\) has almost surely the following properties: a) \(\Lambda\) is \(p\)-Rider, but \(q\)-Rider for no \(q<p\). b) \(\Lambda\) is a \(\Lambda(q)\) set for every \(q\). c) \(\Lambda\) is uniformly distributed; in particular it is dense in the Bohr group and \(c_0\subset C_\Lambda\). d) \(\Lambda\) is not a \(UC\) set.
The new point is \((d)\).

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
42A55 Lacunary series of trigonometric and other functions; Riesz products
42A61 Probabilistic methods for one variable harmonic analysis
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References:

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