Thin sets of integers in Harmonic analysis and p-stable random Fourier series

arXiv:0902.2625v1 [math.FA] 16 Feb 2009 Thin sets of integers in Harmonic analysis and p-stable random Fourier series P. Lef`evre, D. Li, H. Queff´elec, L. Rodr´ıguez-Piazza Dedicated to the memory of Daniel Rider November 4, 2013 Abstract. We investigate the behavior of some thin sets of integers defined through random trigonometric polynomial when one replaces Gaussian or Rademacher variables by p-stable ones, with 1< p < 2. We show that in one case this behavior is essentially the same as in the Gaussian case, whereas in another case, this behavior is entirely different. AMS Classification numbers. Primary: 43A46 ; secondary: 42A55; 42A61; 60G52 Key-words. Fourier series; quasi-independent set; Rider set; stable random variables; stationary set 1 Introduction Let Gbe a compact, abelian group (which will be mostly the circle T), equipped with its normalized Haar measure m, and Γ its (discrete) dual. We will denote by Pthe set of finite sums Pγ∈Γcγγ, i.e. the vector space generated by Γ, and by PΛthe set of finite sums Pγ∈Λcγγ, where Λ is a subset of Γ. We recall ([22]) that Λ is called a Sidon set if, for some constant C, we have the following a priori inequality: kfkF1:= X γ|ˆ f(γ)| ≤ Ckfk∞,∀f∈ PΛ.(1.1) 1 The best constant Cin (1.1) is called the Sidon constant of Λ. A long standing problem, solved in the positive by Drury ([4]) at the beginning of the seventies, was whether the union of two Sidon sets is again a Sidon set. A little after Drury’s result, Rider ([19]) gave the following necessary and sufficient condition for Sidonicity, from which the result becomes obvious: kfkF1≤C[[f]],∀f∈ PΛ.(1.2) Here, we have set: [[f]] = E  Xεγˆ f(γ)γ  ∞,(1.3) where (εγ)γis a sequence of i.i.d. Rademacher random variables, defined on some probability space Ω, i.e. independent and taking the values +1 and −1 with equal probability 1 2, and where Estands for expectation on Ω. This norm was thoroughly studied by Marcus and Pisier ([13]) and is called the Cas-norm in the space of almost surely continuous random Fourier series. These two authors proved in particular the non-trivial fact that one could as well use a standard gaussian sequence instead of a Rademacher one, and obtain an equivalent norm (see [16], Th´eor`eme 7.1). Pisier ([16]) realized that Sidonicity can also be characterized by the a priori inequality: kfk∞≤C[[f]],∀f∈ PΛ.(1.4) This is a general fact, the proof of which we recall for the convenience of the reader, and which motivates the forthcoming definition of stationarity: let (Zγ)γ∈Γbe a collection of i.i.d. copies of a complex-valued, centered and integrable random variable Z, and set, for every trigonometric polynomial f, [[f]]Z=E  XZγˆ f(γ)γ  ∞. We have the following simple proposition. Proposition 1.1. Let Λ⊂Γbe such that kfk∞≤C[[f]]Z,∀f∈ PΛ.(1.5) Then, Λis a Sidon set. Proof. Let ( e Zγ)γ∈Γ) be an independent family, with each e Zγa symmetrization of Zγ. Since the latter variables are centered, we have E  XZγˆ f(γ)γ  ∞≤2E  Xe Zγˆ f(γ)γ  ∞, 2 so we may as well assume the Zγ’s symmetric from the beginning. If εγ=±1, we therefore have X γ εγˆ f(γ)≤  X γ εγˆ f(γ)γ  ∞≤CE  XZγˆ f(γ)γ  ∞. Taking the supremum on all choices of ±1 gives classically ([10], Chapitre 5, Proposition IV.2): X γ|ˆ f(γ)| ≤ DE  XZγˆ f(γ)γ  ∞, with D=π 2C. Now, we truncate our variables Zγat a level M:Zγ=Z′ γ+Z′′ γ, where Z′ γ=Zγ1I{|Zγ|≤M}, Z′′ γ=Zγ1I{|Zγ|>M}; Mbeing adjusted so as to have E|Z′′ γ| ≤ 1 2D. We now see that X γ|ˆ f(γ)| ≤ DE  X γ Z′ γˆ f(γ)γ  ∞+DE  X γ Z′′ γˆ f(γ)γ  ∞ ≤DE  X γ Z′ γˆ f(γ)γ  ∞+DX γ|ˆ f(γ)|E|Z′′ γ| ≤DE  X γ Z′ γˆ f(γ)γ  ∞+1 2X γ|ˆ f(γ)| ≤4MD E  X γ εγˆ f(γ)γ  ∞+1 2X γ|ˆ f(γ)| (here, (εγ)γis a Rademacher sequence, and we used the usual “contraction principle”: see [10], Chapitre 3, Th´eor`eme III.3); whence X γ|ˆ f(γ)| ≤ 8MD E  X γ εγˆ f(γ)γ  ∞. Now, we are in position to apply Rider’s Theorem ([10], Chapitre 5, Th´eor`eme IV.18) to conclude that Λ is Sidon.  Note that Proposition 1.1 has an easy converse (which we state for further reference). Proposition 1.2. If Λis a Sidon set, then k k∞and [[ ]]Zare equivalent norms on PΛ. 3 Proof. Let f∈ PΛand fω Z=PγZγ(ω)b f(γ)γ. On one hand, one has Eωkfk∞≤X γ∈Λ E(|Zγ|)|b f(γ)|=kZ1k1X γ∈Λ|b f(γ)| ≤ CkZk1kfk∞, and, on the other hand, Pγ|Zγ(ω)b f(γ)| ≤ Ckfω Zk∞, since Λ is Sidon and fω Z∈ PΛ; hence, by integrating, PγE(|Zγ|)|b f(γ)| ≤ C[[f]]Z, and kfk∞≤X γ|b f(γ)| ≤ C kZ1k1 [[f]]Z. Pisier ([16]) also studied the subsets Λ of Γ verifying the reverse inequality of (1.5) (see [16], D´efinition 6.2), namely: [[f]] ≤Ckfk∞,∀f∈ PΛ.(1.6) and he called those sets stationary, proving in particular ([16], Proposition 6.2) that the cartesian product of dSidon sets is always stationary (the first named author [7] proved that, for example, {3k1+···+ 3kd; 1 ≤ k1<···< kd}is also a stationary set). Another well-known notion is that of q-Sidonicity, 1 ≤q < 2 ([12], [16], D´efinition 6.1). The subset Λ is called q-Sidon if, for some constant C, we have: kfkFq:= X γ|ˆ f(γ)|q1 q≤Ckfk∞,∀f∈ PΛ.(1.7) After the work of Rider, the following notion was also introduced ([11] and [21]): the subset Λ is called q-Rider if, for some constant C, we have this time: kfkFq≤C[[f]],∀f∈ PΛ.(1.8) It is immediate to see that every q-Sidon set is a q-Rider set, and Rider’s result can be formulated in saying that the converse holds for q= 1. Whether this converse holds for each q∈(1,2) is an open problem, in spite of several non-trivial partial results ([8]). Let us mention that the cartesian product of dinfinite Sidon sets is q-Sidon with q=2d d+1 and not better ([12]). We could of course study those notions for other probability laws than the (subgaussian) Rademacher laws or gaussian ones. This is precisely the 4 aim of this work, where we will be interested in the complex, symmetric, p-stable random variables Z, 1 < p < 2, which can be defined through their characteristic function: EeiRezZ = exp(−|z|p),∀z∈C. The case p= 2 is the gaussian case already studied. The case 1 < p < 2 is in some sense more delicate, because in spite of the nice stability property: N X n=1 anZn∼N X n=1 |an|p1 pZ1 from which those variables borrow their name, their integrability properties are fairly poor: Z∈Lsfor each s < p, but Z6∈ Lp(in fact, Z∈Lp,∞). Yet, this case has also been studied in great detail by Marcus and Pisier in [14], who in particular introduced the following p-stable norm on the space of trigonometric polynomials [[f]]p=E  X γ∈γ Zγˆ f(γ)γ  ∞,∀f∈ P,(1.9) where (Zγ)γis a family of independent copies of a complex p-stable, symmetric, random variables. Observe that this has a meaning, since the Zγ’s are integrable. Moreover, due to a general comparison principle of Jain and Marcus ([5]), one has the following inequality, where the implied constants only depend on p1and p2: 1< p1< p2≤2⇒[[f]]p2≤C(p1, p2) [[f]]p1,∀f∈ P.(1.10) In other terms, the smaller p, the bigger the corresponding [[ ]]p-norm. In particular, those new norms are bigger than the previously mentioned Rademacher and Gaussian norms on P. The questions which we examine in this work are the following: what do the notions of stationarity, q-Riderness, become if we replace the gaussian variables by p-stable ones? After having established, in Section 2, one basic property of the [[ ]]pnorm, namely a lower p-estimate, we prove in Section 3, that a p-stationary set is in fact Sidon (and, of course, conversely) as soon as p < 2, and we study, in Section 4, several equivalent forms of p-stable q-Riderness, and show that this apparently new notion coincides with that of s-Riderness for an appropriate value of the parameter s, depending on pand q. We end with some comments. 5 2 Basic properties of the p-stable norm We will need the following two theorems on p-stable norms. They are more or less straightforward consequences of a basic result of Marcus and Pisier. First, we introduce some notation: Fpwill denote the set of functions f∈L2=L2(G, m) such that their Fourier transform is in ℓp=ℓp(Γ), equipped with the norm kfkFp:= kˆ fkp, which we already encountered in Section 1 (see (1.7)). We shall denote by k.kψthe Luxemburg norm in the Orlicz space associated to an Orlicz function ψ. Let r > 0, we shall be mainly interested in the Orlicz function ϕr, where ϕr(x) = x1 + log(1 + x)1 r, and the conjugate Orlicz function ψr, where ψr(x) = exr−1. Finally let A(p, ϕp′) (where p′is the conjugate exponent of p) be the space of all functions in L2(G) which can be written as f= ∞ X n=1 hn∗kn, with: ∞ X n=1 khnkFpkknkϕp′<∞, and equipped with the norm kfkA(p,ϕp′)= inf n∞ X n=1 khnkFpkknkϕp′o, where the infimum runs over all possible representations of f. 6 With those notations, the basic result alluded to above stands as follows, under a simplified form which will be sufficient for us ([14], Theorem 5.1): Theorem 2.1 (Marcus-Pisier). The norms [[f]]pand kfkA(p,ϕp′)are equivalent on the space Pof trigonometric polynomials on G. Two important consequences of that theorem, which are not explicited in [14] in the p-stable case, are the following (see [16], Proposition 7.1 for the case p= 2). Theorem 2.2 (Contraction principle for the p-stable norm). Let f∈ Pand (ξγ)γbe a collection of functions in Lp(0,1), bounded in Lp. Denote the integral over (0,1) by E′. Then, we have for some positive constant a: E′hhX γ ξγˆ f(γ)γiip p1 p≤asup γkξγkLp(0,1) [[f]]p.(2.1) Proof. Let f=P∞ n=1 hn∗knbe an admissible decomposition of f, and let ω′∈(0,1), as well as fω′=Pγξγ(ω′)ˆ f(γ)γ. We can write fω′= P∞ n=1 Hn(ω′)∗kn, with Hn(ω′)∈Fpand \ Hn(ω′)(γ) = ξγ(ω′)ˆ hn(γ). Set for convenience X(ω′) = [[fω′]]pand Yn(ω′) = kHn(ω′)kFp. We see that X(ω′)≤ ∞ X n=1 [[Hn(ω′)∗kn]]p≤a ∞ X n=1 Yn(ω′)kknkϕp′, where ais some constant given by the Marcus-Pisier Theorem above. Now taking Lp-norms in Lp(0,1) and using the triangle inequality, we get: kXkLp(0,1) ≤a ∞ X n=1 kYnkLp(0,1)kknkϕp′≤aC ∞ X n=1 khnkFpkknkϕp′, where C= supγkξγkLp(0,1). Taking the infimum over all possible representations of fgives us the result, possibly changing the constant a. The second basic consequence is: 7 Theorem 2.3 (Lower p-estimate for the [[ ]]p-norm).Let f, f1,...,fN be trigonometric polynomials such that: |ˆ f(γ)| ≥ N X j=1 |ˆ fj(γ)|p1 p,∀γ∈Γ. Then, the constant abeing as in (2.1): [[f]]p≥a−1N X j=1 [[fj]]p p1 p·(2.2) Proof. Let A1, . . . , ANbe a partition of (0,1) in sets of Lebesgue measure 1/N and, for each γ∈Γ, ξγ∈Lp(0,1) be defined by: ξγ=N1 p N X j=1 ˆ fj(γ) ˆ f(γ)1IAj. It is clear that kξγkLp(0,1) ≤1 by our assumption, and by definition we have E′hhX γ ξγˆ f(γ)γiip p= N X j=1 hhX γ ˆ fj(γ) ˆ f(γ) ˆ f(γ)γiip p= N X j=1 [[fj]]p p, so that an application of (2.1) gives the result.  We will end this section with the following estimate. This is undoubtedly known, but we did not find any explicit mention; so we are going to give some words of explanation. Lemma 2.4. Let λ1< λ2<···< λn∈N, with λn≥2. If f(t) = Pn j=1 eiλjt, t∈T, one has, for some constant C > 0: [[f]]p≤C n1/p(log λn)1/p′. Proof. From (4.6) of [14] (or Remark 1.7, page 186 of [15]), there is a constant K > 0 such that [[g]]p≤K ∞ X k=2 1 k(log k)1/p ∞ X j=k|bg(j)|p1/p 8 for every trigonometric polynomial gwith spectrum in N. Here one has P∞ j=k|b f(j)|p≤nfor k≤λnand P∞ j=k|b f(j)|p= 0 for k > λn; hence [[f]]p≤K n1/p λn X k=2 1 k(log k)1/p ≤C n1/p(log λn)1/p′. 3p-stable stationary sets are Sidon for p < 2 The aim of this Section is to prove the following: Theorem 3.1. Let Λ⊂Γbe a p-stationary set (1< p ≤2), i.e. a set satisfying the following inequality, for some constant C=Cp: [[f]]p≤Ckfk∞,∀f∈ PΛ. Then, if p < 2,Λis a Sidon set. The difference between the cases 1 < p < 2 and p= 2 (for which non- Sidon stationary sets exist), comes from the fact the p-stable norm for p < 2 is bigger than the usual Pisier norm; hence having an upper estimate for it on some space forces the smallness of this space. In order to prove Theorem 3.1, we will need the following simple lemma : Lemma 3.2. If Λ⊂Γis a p-stationary set, the norms [[ ]]2and [[ ]]pare equivalent on the space PΛ. Proof of Lemma 3.2. Let f∈ PΛ, (εγ)γa Rademacher sequence, and fω=Pγεγ(ω)ˆ f(γ)γ. By symmetry, we have [[f]]p= [[fω]]p≤Ckfωk∞, where Cis the stationarity constant of Λ. Integrating over ωgives: [[f]]p≤ CEkfωk∞=C[[f]]2, which finishes the proof, since we know from (1.10) that the reverse inequality always holds (we wrote [[ ]]2instead of [[ ]] to make a clear distinction between the subgaussian (i.e. Rademacher) or gaussian case, and the p-stable one).  Proof of Theorem 3.1. From the previous Lemma 3.2, it will be enough to show that, if Λ is not Sidon, then the norms [[ ]]2and [[ ]]pare not equivalent on PΛ. It will be convenient to introduce first some notation. Recall that a 9 isomorphic to M(Fp, Lψp′), the set of multipliers from Fpto Lψp′. This result (well-known for p= 2: see [13]) follows in a standard way from the delicate Theorem 2.1, as it is already the case for p= 2 and we will not detail that formal extension. 5. Once and for all, we set ε=p−q q(p−1) = 1 −p′ q′·(4.4) 1 α=1 p+1 q′(4.5) β=ε p′+1 p−1 2=1 q−1 2·(4.6) We first prove the following simple proposition (the case q= 2 being already known [11]), which will actually follow from Theorem 4.3 below, but which will motivate this theorem. Proposition 4.1 (Mesh condition). Let Λ∈Zbe a p-stable q-Rider set. Then, there exists a constant Ksuch that, for each integer N≥2, one has : |Λ∩{1,...,N}| ≤ K(log N)(p−1)q p−q=K(log N)1 ε.(4.7) Proof. Set B= Λ ∩ {1,...,N}={λ1<···< λn}, and f=Pn j=1 eλj. The assumption and Lemma 2.4 give us, for some constant C, that n1 q≤ C[[f]]p≤CC0n1 p(log N)1 p′. Grouping terms gives n≤K(log N)(p−1)q p−q, where we set K= (CC0)pq p−q, proving our claim.  We will now prove the main result of this section: Theorem 4.2. Let Λ⊂Γ. Λis a p-stable q-Rider set if and only if Λis an s-Rider set, with s=2q′ 2q′−p′· Actually we are going to prove the following more precise theorem: 16 Theorem 4.3. Let Λ⊂Γ. Then, the following conditions are equivalent: (1) Λ is a p-stable q-Rider set; (2) ℓq′(Λ) ֒→ M(Fp, Lψp′); (3) ℓα(Λ) ֒→Lψp′; (4) ψp′(A)≤C|A|1/α for some constant C > 0, and every finite subset A of Λ; (5) q(A)≥c|A|ε, for some constant c > 0, and for all finite subsets {0} 6=A⊂Λ; (6) Λ is an s-Rider set, with s=2q′ 2q′−p′=2q(p−1) p−2q+pq · Point out the simple relation: 2q′=s′p′. Moreover, to precise the behavior for the “degenerate” cases: when q= 1, we have s= 1 as well (remember Proposition 1.1). On the other hand, when s= 1, q= 1 and any pfits. Let us make some comments. This result is known for p= 2 and has been proved by the fourth-named author ([20], [21], Teorema III.2.3). The symbol ֒→means that the left-hand space is mapped to the right-hand one by means of the Fourier transform or of its inverse. Recall that q(A) denotes the largest possible cardinality of a quasi-independent subset of A, and that the definition of quasi-independent sets is given at the beginning of the proof of Theorem 3.1. Remark 1. We know (see [11]) that the mesh condition for s-Rider reads as |Λ∩{1,...,N}| ≤ K(log N)s 2−s. But, s 2−s=1 ε, and we fall again on (4.7) of Proposition 4.1. Remark 2. The preceding theorem can be read in two ways. First, any p-stable q-Rider set is actually an s-Rider set for the right value of s. But on the other hand, if one fixes some s∈(1,2) and a set Λ which is an s-Rider set, one can choose either por qin order to realize Λ as a p-stable q-Rider set. Nevertheless, one has to be careful. Let us precise this: a. If one fixes p∈(1,2], then one can choose qsuch that q′=s′p′/2 and Λ is a p-stable q-Rider set. Point out that q < p ≤2. b. If one fixes q∈(1,2), then one can choose p∈(1,2] such that p′=2q′ s′if and only if we have s≥q. In that case, Λ is a p-stable q-Rider set. 17 Proof of Theorem 4.3. (1) ⇔(2). The Fourier transform maps XΛto ℓq(Λ) if and only if its transpose maps ℓq′(Λ) to the dual of XΛ. The result now easily follows from the previous description of the dual of X. (2) ⇒(3). Let f=X γ∈Λ cγγ∈ PΛ. We can write (in short: see (4.5), we have ℓα=ℓp. ℓq′): cγ=aγbγ, where |aγ|=|cγ|1−θ,|bγ|=|cγ|θ, with θ=α p= 1 −α q′and kakq′=kck1−θ α,kbkp=kckθ α. If we set P=Pγbγγ, we have from (2) that kfkψp′≤Ckakq′kPkFp, where Cis some constant. Equivalently: kfkψp′≤Ckck1−θ+θ α=Ckckα, which was our claim. (3) ⇒(2). This is obvious since 1 α=1 p+1 q′· (3) ⇒(4). Indeed, ψp′(A)≤Ckc 1IAkα=C|A|1 α. (4) ⇒(5). We use a result that we proved in ([9], Proposition 3.2), namely that, for any finite set A⊂Γ, A6={0}, and any r > 0, we have: q(A)≥Cr|A| ψr(A)r(4.8) We use (4.8) with r=p′. Our assumption implies that q(A)≥C|A|(1−1 α)p′, and 1−1 αp′=1−1 q′−1 pp′=1 p′−1 q′p′= 1 −p′ q′=ε. (5) ⇒(6). By [20] or [21], Teorema III.2.3, we know that this condition is equivalent to the fact that Λ is an s-Rider set where ε=2 s−1. As ε= 1−p′ q′, we have s=2q′ 2q′−p′. (6) ⇒(3). Either using [20], [21], Teorema III.2.3 or applying the equivalence of (1) and (3) when p= 2 and sinstead of q, we know that ℓ˜α(Λ) ֒→Lψ2 where 1 ˜α=1 2+1 s′. On the other hand, we obviously have ℓ1(Λ) ֒→L∞. As 18 p > q, we have ˜α > α, so a standard interpolation argument implies that ℓα(Λ) ֒→Lψrwith 1 α=θ ˜α+1−θ 1and 1 r=θ 2+1−θ ∞ We obtain θ=1 q′−1 p′1 s′−1 2−1=2 p′, since s′p′= 2q′. Hence r=2 θ=p′.  The following theorem has two aims. First the proof of the preceding theorem is not self-contained: to prove that (5) ⇒(6), we used the fact that the theorem was already known for p= 2. The following theorem provides a proof of this result as well. On the other hand, this will actually provide a stronger result, which cannot a priori be obtained just using the case p= 2 (i.e. assuming the results of [20] or [21]). Nevertheless, the proof proceeds as in [21], using a difficult lemma of Bourgain on quasi-independent sets. Though there is essentially no change with regard to [21], we will give the details, for the convenience of the reader. The links beetween the values of the parameters are the same as before. Theorem 4.4. Let Λ⊂Γ. The following conditions are equivalent: (i)q(A)≥c|A|ε, for some constant c > 0, and for all finite subsets {0} 6=A⊂Λ; (ii)Cp−as Λ:= XΛ֒→ℓq,1(Λ); (iii) Λ is a p-stable q-Rider set; (iv)Cp−as Λ:= XΛ֒→ℓq,∞(Λ). Recall that ℓq,1=ℓq,1(Λ) (resp. ℓq,∞=ℓq,∞(Λ)) is the Lorentz space of families a= (aγ)γ∈Λtending to 0 whose decreasing rearrangement (a∗ n)n satisfies X n≥1 a∗ n n1/q′<∞(resp. sup n≥1 n1/qa∗ n<∞). One has ℓq,1(Λ) ֒→ℓq(Λ) ֒→ℓq,∞(Λ). Proof. The implications (ii)⇒(iii)⇒(iv) are obvious. (iv)⇒(i) is easy with help of Theorem 3.5: for any finite subset Aof Λ, we have q(A)≥K−1[[A]]p |A|1/p p′ ≥K1|A|1/q−1/pp′ 19 which gives (iv), since p′1 q−1 p=p′1 p′−1 q′=ε. It remains to prove the difficult part ((i)⇒(ii)) of this theorem. We first need the following lemma. Lemma 4.5 ([21], Lema III.2.6).Let Λbe a set satisfying the condition (5) of Theorem 4.3. For any finite subset Aof Λsuch that A6={0}and c|A|ε≥2, there exist Npairwise disjoint quasi-independent sets B1,...,BN⊂Asuch that: a)2 c|A|1−ε≥N≥1 2c|A|1−ε; b)c|A|ε≥ |Bj| ≥ c 2|A|εfor every j= 1,...,N. Proof of Lemma 4.5. By our assumption, there is a quasi-independent subset Bof Asuch that |B| ≥ c|A|ε. Since c 2|A|ε≥1, we can find a (quasiindependent) subset B1of Bsuch that c 2|A|ε≤ |B1| ≤ c|A|ε. Assume now that pairwise disjoint quasi-independent subsets B1,...,Bn of Ahas been constructed such that c|A|ε≥ |Bj| ≥ c 2|A|εfor 1 ≤j≤n. There are two possibilities: (i) If |A|/2>Sn j=1 Bj, we choose in A\Sn j=1 Bja quasi-independent subset Bn+1 whose cardinal is |Bn+1| ≥ cA\ n [ j=1 Bjε≥c|A| 2ε≥c 2|A|ε, and which we can also choose such that |Bn+1| ≤ c|A|ε. (ii) If |A|/2≤Sn j=1 Bj, we stop the process at N=n. Indeed, we then have: |A| 2≤ N [ j=1 Bj≤N c |A|ε and hence N≥1 2c|A|1−ε. On the other hand, since B1,...,BNare disjoint, one has |A| ≥  N [ j=1 Bj= N X j=1 |Bj| ≥ Nc 2|A|ε; hence N≤2 c|A|1−ε. That ends the proof of Lemma 4.5.  Now, we have to show that there exists a constant C > 0 such that kb fkq,1≤C[[f]]p 20 for every trigonometric polynomial f∈ PΛ. By homogeneity, we may assume that kˆ fk∞= 1, and we consider the level sets Aj={γ∈Λ ; 2−j+1 ≥ |b f(γ)|>2−j}, j = 1,2,... Recall now the following result of Bourgain ([1], Lemma 2; see also [2], and [10], Chapitre 12, Lemme I.10). Lemma 4.6 (Bourgain). There exists a numerical constant R > 10 such that for every family B1,...,BLof pairwise disjoint finite quasi-independent sets such that |Bl+1| |Bl|≥R , for l= 1,...,L−1, one can find, for each l= 1,...,L, a subset Cl⊂Blsuch that: |Cl| ≥ 1 10 |Bl|,for all l= 1,...,L, and the union SL l=1 Clis quasi-independent. Setting R1= max{R1/2β,(2R)ε}, where Ris the above constant and βis given by (4.6), we define j1= 1 and jl+1 = min{j > jl;|Aj|> R1|Ajl|},(4.9) whenever this last set is nonempty; we stop and take L=lwhen it is empty (this eventually happens since fis a trigonometric polynomial). Set Nl=|Ajl|. We have the following upper estimate: kb fkq,1≤X j≥1 2−j+1k1IAjkq,1≤ L X l=1 X jl≤j<jl+1 2−j+1q|Aj|1/q , since k1IAkq,1= |A| X n=1 1 n1/q′≤Z|A| 0 dx x1/q′=q|A|1/q. 21 Now, we have |Aj| ≤ R1|Ajl|for jl≤j < jl+1; hence we get kb fkq,1≤q L X l=1 R1/q 12−jl|Ajl|1/q X jl≤j<jl+1 2−j+1+jl ≤4qR1/q 1 L X l=1 2−jlN1/q l.(4.10) We are now going to use Lemma 4.5. For this purpose, let us denote by Tthe first index lsuch that c N2β l≥2. If no such an index exists, we will set T=L+ 1. We will split the sum (4.10) into two parts. First, the Cauchy-Schwarz inequality gives T−1 X l=1 2−jlN1/q l≤T−1 X l=1 2−2jlNl1/2T−1 X l=1 N2β l1/2 ≤X γ∈Λ|b f(γ)|21/2T−1 X l=1 NT−1 RT−l−1 12β1/2 =kfk2Nβ T−1X k≥0 R−2kβ 11/2. Since N2β T−1<2/c and R2β 1≥R > 10 >4, we get T−1 X l=1 2−jlN1/q l≤ kfk22 c1/24 31/2 and T−1 X l=1 2−jlN1/q l≤2 √c[[f]]p.(4.11) For l≥T, we apply Lemma 4.5 to the set Ajl: we get Mlpairwise disjoint quasi-independent subsets Bl,1,...,Bl,Ml⊂Ajlsuch that c 2Nε l≤ |Bl,m| ≤ cNε l, m = 1,...,Ml,(4.12) 22 with 1 2cN1−ε l≤Ml≤2 cN1−ε l, l =T, . . . , L. (4.13) Since Nl+1 ≥R1Nl, we get, for T≤l < L: ML Ml≥R1−ε 1 4≥1 4(4.14) and |Bl+1,m| |Bl,m′|≥Rε 1 2≥R. (4.15) Using (4.14), we can find, for each l=T,...,L−1, a map φl:{1,...,ML} → {1,...,Ml} such that |φ−1 l(m)| ≤ 4ML Ml, , m = 1,...,Ml. Applying, for each m= 1,...,ML, Lemma 4.6 to the sequence BT,φT(m),...,BL−1,φL−1(m), BL,m , thanks to (4.15), we get, for each l=T,...,L and each m= 1,...,ML, a quasi-independent set Cl,m such that |Cl,m|>1 10 |Bl,φl(m)| ≥ c 20 Nε l, l =T, . . . , L, (4.16) and such that L [ l=T Cl,m is quasi-independent for m= 1,...,ML.(4.17) We now introduce, for m= 1,...,ML, the trigonometric polynomial gm= L X l=TMl 4ML1/p2−jlX γ∈Cl,m γ. We claim that, for every γ∈Γ, one has |b f(γ)|p≥ ML X m=1 |bgm(γ)|p.(4.18) 23 Indeed, it suffices to check that for γin the spectra of the gm’s, and if γis in some set Bl,m0, it cannot be in another one, because these sets are pairwise disjoint; hence ML X m=1 |bgm(γ)|p= 2−pjlMl 4ML{m;γ∈Cl,m}≤2−pjlMl 4ML|φ−1 l(m0)| ≤2−pjl≤ |b f(γ)|p. It follows from the lower p-estimate of the norm [[ ]]p(Theorem 2.3) that a[[f]]p≥ML X m=1 [[gm]]p1/p.(4.19) But, by (4.17), the spectrum of gmis quasi-independent, and every quasiindependent set is a Sidon set, with constant ≤5 (see the beginning of the proof of Theorem 3.1), so kbgmk1≤5kgmk∞, and kbgmk1≤5 [[gm]]p, by Proposition 1.2. It follows, from (4.13) and (4.16), that: 5 [[gm]]p≥ L X l=T|Cl,m|Ml 4ML1/p2−jl≥1 41/pM1/p L L X l=T c 20 Nε lN1−ε l 2c1/p2−jl =c1/p′ 20(8ML)1/p L X l=T Nε+1−ε p l2−jl. Since ε+1−ε p=1−1 pε+1 p=ε p′+1 p=1 q−1 p+1 p=1 q, we get [[gm]]p≥c1/p′ 100(8ML)1/p L X l=T N1/q l2−jl.(4.20) Therefore (4.19) gives [[f]]p≥c1/p′ 100a(8ML)1/p ML X m=1 L X l=T N1/q l2−jlp1/p =c1/p′ 100a.81/p L X l=T N1/q l2−jl.(4.21) 24 Putting (4.21) together with (4.11), we get: L X l=1 N1/q l2−jl≤2 √c+100a.81/p c1/p′[[f]]p. It remains to use (4.10) to obtain: kb fkq,1≤4qR1/q 12 √c+100a.81/p c1/p′[[f]]p, and achieve the proof of Theorem 4.4.  5 The link with Orlicz spaces We are going to characterize s-Rider sets in terms of continuous mapping to Orlicz spaces (remember the beginning of Section 2). Theorem 5.1. Let Λ⊂Γ,s∈(1,2) and rbe greater than both 2and ρ=2−s s−1. Let ˜p=2r 2r−ρ· The following conditions are equivalent: (i) Λ is an s-Rider set; (ii)C˜p−as Λ֒→Lψr; (iii)For every finite subset Aof Λ, we have ψr(A)≤C[[A]]˜p, where C does not depend on A. Proof. (i)⇒(ii). We already know that we can realize Λ as a p-stable q-Rider set with p=r′≤2. Then the value of qis fixed by the relation q′=rs′/2. By (3) of Theorem 4.3, we know that ℓα(Λ) ֒→Lψrwith 1 α= 1 p+1 q′= 1 −2−s sr . Let us point out that α < 2. Now we can use Theorem 4.3 again to realize Λ as a ˜p-stable α-Rider set but only (see Remark 2 after that theorem) when s≥α. This condition is fulfilled since it is equivalent to the condition r≥ρ. Moreover the value of ˜pis fixed by the relation ˜p′=2α′ s′=2r 2r−ρ·The conclusion follows. (ii)⇒(iii). This is obvious. (iii)⇒(i). Fix any finite subset Aof Λ. We shall use ε=2 s−1· 25