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Fractional cartesian products of sets

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Fractional cartesian products of sets ANNALES DE L’INSTITUT FOURIER RONC. BLEI Fractional cartesian products of sets Annales de l’institut Fourier, tome 29, no 2 (1979), p. 79-105. <> © Annales de l’institut Fourier, 1979, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » ( implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Ann. Inst. Fourier, Grenoble 29, 2 (1979), 79-105. FRACTIONAL CARTESIAN PRODUCTS OF SETS by Ron C. BLEI (*) N-fold sums of «independent» sets serve in harmonic analysis as prototypical examples of 2N/(N + l)-Sidon sets, and A(q) sets whose \(q) constants' growth is (9 (^N/2). Moreover, these features are exact : N-fold sums of independent sets are not (2N/(N - h i ) — e)-Sidon and to not have \(q) constants' growth asymptotic to qW2-^, for any e > 0 (see [4], [6] and [2]). In this paper, given any number p e (1, 2), we display a set that is p-Sidon but not(p — e)-Sidon for any e > 0. The same pool of examples contains, for any number a e [1/2, oo), a set whose A(q} constants' growth is ^(q") but not ^W -e) ^or anv £ > 0. This answers questions raised in [4] and [6], and a question that is implicit in [2]. The type of sets displayed here exhibits « combinatorial » and « analytic » properties that one would expect« fractio- nal » cartesian products (sums) of sets to possess, and hence the title of the paper. This class of sets naturally arises in the study of multidimensional extensions of Grothendieck's inequality ([!]); it is that study that led to the present work. 1. Definition

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