Some new thin sets of integers in harmonic analysis

a Xi :0912.4214 1 [ma h.FA] 21 Dec 2009 Some new hin se s o in ege s in Ha moni Analysis Daniel Li, He é Queéle, Luis Rod íguez-Piazza Abs a . We andomly ons u a ious subse s Λ o he in ege s whih ha e bo h smal lness and la geness p ope ies. They a e smal l sine hey a e e y lose, in a ious meanings, o Sidon se s: he on inuous un ions wi h spe um in Λ ha e uni o mly on e gen se ies, and hei Fou ie oeien s a e in ℓp o al l p > 1 ; mo eo e , al l he Lebesgue spaes Lq Λ a e equal o q < +∞ . On he o he hand, hey a e la ge in he sense ha hey a e dense in he Boh g oup and ha he spae o he bounded un ions wi h spe um in Λ is non sepa able. So hese se s a e e y die en om he hin se s o in ege s p e iously known. Résumé . On ons ui aléa oi emen des ensembles Λ d'en ie s posi i s jouissan si- mul anémen de p op ié és qui les on appa aî e à la ois omme pe i s e omme g ands. Ils son pe i s a ès p ohes à plus d'un éga d des ensembles de Sidon: les on ions on inues à spe e dans Λ on une sé ie de Fou ie uni o mémen on e - gen e, e on des oeien s de Fou ie dans ℓp pou ou p > 1 ; de plus, ous les espaes de Lebesgue Lq Λ oïniden pou q < +∞ . Mais ils son pa ail leu s g ands au sens où ils son denses dans le ompa ié de Boh e où l'espae des on ions bo nées à spe e dans Λ n'es pas sépa able. Ces ensembles son don ès dié en s des ensembles mines d'en ie s onnus aupa a an . Key-wo ds . e go di se  launa y se  Λ(q) -se  quasi-indep enden se  andom se  p -Ride se  Rosen hal se  p -Sidon se  se o uni o m on e gene  uni o mly dis ibu ed se . Ma hema is Sub je Classia ion . P ima y : 42A36  42A44  42A55  42A61  43A46; Seonda y : 60D05 In o du ion I is well known ha he Fou ie se ies o an in eg able un ion dened on he uni -i le T=R/2πZ o he omplex plane C an b e badly b eha ed. Fo example, i is well known ha he e exis on inuous un ions whose Fou ie se ies is no e e ywhe e on e gen (see [30℄, Th. 18.1, and Th. 19.5 o he op imal esul ), and in eg able ones wi h e e ywhe e di e gen Fou ie se ies (see [30℄, Th. 19.2 o ins ane; see also [29℄). The p oblem o hin se s o in ege s is he ollowing: ins ead o onside ing all he in eg able un ions on T , o all he on inuous ones, we onside only 1 hose whose sp e um ( he se whe e hei Fou ie o eien s do no anish) is on ained in a p es ib ed subse Λ o he in ege s Z . This se Λ will b e said  hin i he Fou ie se ies o hese un ions b eha es b e e han in he gene al ase. A ypial example is Λ = {1,3,32,...,3n,...} . I is well known (see [62℄, o ins ane) ha e e y in eg able un ion wi h sp e um in Λ ( ∈L1 Λ ) is a ually squa e in eg able, and ha e e y on inuous un ion wi h sp e um in Λ ( ∈ CΛ ) has a no mally on e gen Fou ie se ies (equi alen ly b ∈ℓ1 ). In his seminal pap e [54℄, W. Rudin dened wo no ions o hinness o Λ : Λ is a Sidon se i ∈ CΛ implies ha b ∈ℓ1 , and Λ is a Λ(q) -se o some q > 1 , i ∈L1 Λ implies ha ∈Lq . These onep s may as well b e dened in he mo e gene al se ing o a ompa ab elian g oup G equipp ed wi h i s no malized Haa measu e, and o a subse Λ o i s dis e e dual g oup Γ . W. Rudin s udied he gene al p op e ies o hose se s and he onne ion b e ween he wo no ions. In pa iula , he showed ha Sidon se s a e Λ(q) -se s o all q < +∞ , and ha , mo e p eisely: (0.1) Λ Sidon implies k kq≤C√qk k2 o e e y Λ -p olynomial and o e e y q≥2 , whe e C is a ons an whih depends only on he Sidon ons an o Λ . Sine hen, se e al new no ions o hin se s eme ged. These inlude p -Sidon se s (see [2℄, [3℄, [4℄, [5℄, [14℄, [18℄, [21℄, [23℄, [24℄, [34℄, [47℄, [60℄), and se s o uni o m on e gene (see [1℄, [19℄, [20℄, [21℄, [26℄, [33℄, [44℄, [45℄, [56℄, [57℄): e e y on inuous un ion wi h spe um in suh a se has i s Fou ie se iesin ℓp o uni o mly on e gen , esp e i ely. Bu he examples o suh se s we e always nea ly he same: p o du s (some imes  a ional p o du s: [3℄, [4℄, [5℄), o sums o Sidon se s, whih is a se e e es i ion o he geome y o he Banah spae CΛ . Fo example, F. Lus Piqua d ([40℄) p o ed ha : (0.2) The inje i e enso p o du ℓ1b ⊗ε···b ⊗εℓ1 has he Shu p op e y ( i.e. weakly null sequenes on e ge in no m o ze o). I ollows easily ha : (0.3) I Λ = E1×···×Ek , whe e he Ej 's a e Sidon se s, hen CΛ has he Shu p op e y; in pa iula , CΛ do es no on ain c0 , he spae o sequenes going o ze o a inni y. Sine hese se s we e essen ially he only known examples o p -Sidon se s ( hey a e exa ly 2N/(N+ 1) -Sidon), one ould b elie e ha all p -Sidon se s ha e his p op e y. I should b e men ionned ha in [3℄, R. Blei ons u ed o eah p∈]1,2[ , exa ly p -Sidon se s, using a ional p o du s, so o a die en yp e, bu he o esp onding spae CΛ app ea s as an ℓ1 -sum o ni e dimensional spaes, and so do es ha e he Shu p op e y (we hank R. Blei o his ema k). Beause o his lak o examples, he ompa ison be ween wo lasses o hin se s p o ed o be e y diul : whe he a p -Sidon, o a se o uni o m on e gene is a Λ(q) -se o some q > 1 is s ill an op en p oblem. On he o he 2 hand, onside able p og ess one ning he Sidon se s o Λ(q) -se s has b een made: o example, G. Pisie ([47℄, Th. 6.2) p o ed ha he on e se o (0.1) is ue, and J. Bou gain ([12℄) p o ed ha o eah q > 2 he e exis exa ly Λ(q) -se s, i.e. se s whih a e Λ(q) , bu Λ(q′) o no q′> q . Bo h au ho s used andom me ho ds, and mo e sp eially, J. Bou gain p opula ized he me ho d o sele o s o p o due se e al hin se s Λ wi h unusual p op e ies, suh as b eing uni o mly dis ibu ed, whih implies, by a esul o F. Lus Piqua d ([42℄), ha CΛ on ains c0 and he e o e is no a Rosen hal se ( i.e. he e a e b ounded measu able un ions wi h sp e um in Λ wih a e no almos e e ywhe e equal o a on inuous un ion), and whih also implies ha Λ is dense in he Boh g oup (see [6℄, Theo em 1). This allowed he  s named au ho o see ha he e a e se s o in ege s whih a e Λ(q) o all q < +∞ bu no Rosen hal ([37℄; see also [43℄). The aim o his pap e is he ons u ion o andom se s Λ o in ege s whih ha e hinness p op e ies, bu wih a e no Rosen hal se s ( i.e. CΛ is no he whole L∞ Λ ), a ually suh ha CΛ on ains c0 , and a e dense in he Boh g oup. In iew o (0.3), hese se s will neessa ily b e e y exo i ompa ed o he p e iously known examples. This shows ha eplaing absolu e on e gene o he Fou ie se ies by uni o m on e gene (se s o uni o m on e gene) o by ℓp on e gene o p > 1 ( p -Sidon se s) gi es se s whih a e e y a om Sidon se s. This ons as s wi h Pisie 's esul saying ha Λ is neessa ily a Sidon se whene e b ∈ℓ1,∞ o e e y ∈ CΛ ( om [48℄, Théo ème 2.3 ( i), and he op o page 688). On he o he hand, hough non-Sidon Rosen hal se s do exis ([53℄), i ollows om Bou gain-Milman's o ype heo em ([13℄) ha , o e e y non-Sidon se Λ , CΛ do es on ain ℓn ∞ uni o mly, so ha he p esene o c0 inside CΛ o non-Sidon Λ may app ea no so su p ising. Al hough i is no known whe he Sidon se s may be dense in he Boh g oup, we ob ain in his pape , as men ioned ab o e, se s whih a e dense in he Boh g oup, and a e o uni o m on e gene and p -Sidon o e e y p > 1 . We ons u essen ially ou ypes o se s. Eah o hem will b e a non Rosen hal se , bu a se o uni o m on e gene, Λ(q) o all q < +∞ , and wi h mo eo e addi ional p op e ies o p -Sidonii y. The  s one (Theo em 2.2) is a e y launa y se Λ wi h he nies p op- e ies: i is p -Sidon o all p > 1 . The seond and hi d ones (Theo em 2.5 and Theo em 2.6) a e medium launa y se s: o eah p wi h 1< p < 4/3 , hey a e, in Theo em 2.5, p -Ride (a weake p op e y han b eing p -Sidon, see he deni- ion b elow), bu no q -Ride o q < p , and a e q -Sidon o e e y q > p/(2 −p) ; and in Theo em 2.6, hey a e q -Ride o e e y q > p , bu no p -Ride , and hey a e q -Sidon o e e y q > p/(2 −p) . Finally, he ou h ype (Theo em 2.7) is a se Λ whih is, in some sense as li le launa y as possible i we wan i s ae on eah in e al [N, 2N[ o ha e a b ounded Sidon ons an . I leads o se s whih a e 4/3-Ride , bu no q -Ride o q < 4/3 . We ons u hese se s by using a ious hoies o sele o s, and adding a i hme ial, un ional o p obabilis i a gumen s. The ea men o he las ase equi es a die en p obabilis i app oah, aken om [8℄. 3 I should b e no ed ha in he wo  s ases he se s a e uni o mly dis- ibu ed; in he ou h ase , howe e , he se s Λ only ha e p osi i e upp e densi y in uni o mly dis ibu ed se s. Ne e heless, CΛ s ill on ains c0 , by a esul o F. Lus -Piqua d ([42℄, Th. 5). Aknowledgemen . Pa o his pap e was made when he  s named au ho was a gues o he Depa amen o de Análisis Ma emá io de la Uni e sidad de Se illa in Ap il 1999, and when he hi d named au ho was a gues o he Uni e si é d'A ois in Lens in june 1999. 1 No a ion, deni ions and p elimina y esul s We deno e by T he ompa ab elian g oup o omplex numb e s o mo dulus one, equipp ed wi h i s no malized Haa measu e m . C(T) deno es he spae o on inuous omplex un ions dened on T , equipp ed wi h i s sup no m kk∞ and iden ied as usual wi h he spae o on inuous 2π -p e io di omplex un ions dened on R . I Λ is a subse o he dual g oup Z , CΛ will deno e he subspae o C(T) onsis ing o un ions whose sp e um lies in Λ : b (n)≡ZT e−ndm = 0 i n∈Z Λ, whe e en(z) = zn , o equi alen ly, en( ) = ein . CΛ is he uni o m losu e o he spae PΛ o igonome i p olynomials wi h sp e um in Λ , i.e. he uni o m losu e o he subspae PΛ gene a ed by he ha a e s en , wi h n∈Λ . Fo ∈ C(T) , 1≤q < +∞ , M and N p osi i e in ege s, we shall deno e he Fou ie sums o by: SM,N ( ) = N X −Mb (n)en and he symme i Fou ie sums o by: SN( ) = SN,N ( ) = N X −Nb (n)en. |A| deno es he a dinali y o he ni e se A . A ela ion in Λ⊆Z∗≡Z {0} is a (+1,−1,0) - alued sequene (θk)k∈Λ suh ha P|θk|<+∞ and Pθkk= 0 . The se S={k;θk6= 0} is alled he suppo o he he ela ion, and |S|=P|θk| is alled i s leng h . The ela ion (θ′ k)k∈Λ is said o b e longe han he ela ion (θk)k∈Λ i θk6= 0 implies θk=θ′ k . The se Λ⊆Z∗ is quasi-independen i i on ains no non- i ial ela ion ( i.e. wi h non-emp y supp o ). Typially, Λ = {1,2,4,...,2n,...} is quasi- indep enden . The quasi-indep enden se s a e he p o o ype o Sidon se s, i.e. 4 o se s Λ o whih: kb k1≤Kk k∞ o all ∈ CΛ . The b es ons an K in his inequali y is alled he Sidon ons an o Λ and is deno ed by S(Λ) . We will e e o [39℄ o s anda d no ions on Sidon se s. I is known ha quasi-independen se s a e no only Sidon se s bu hei Sidon ons an is bounded by an absolu e ons an : his ollows om [54℄, Th. 2.4 and [49℄, Lemma 1.7. O he p o o s an b e ound in [48℄, lemme 3.2, and in [9℄, P op. 1. We shall use he a ha S(Λ) ≤8 i Λ is quasi-indep enden . Le us eall now some lassial deni ions and esul s. A se Λ⊆Z is said o b e a Λ(q) -se (whe e q > 2 ) i he e exis s a p osi i e ons an Cq suh ha k kq≤Cqk k2 o e e y ∈ PΛ . The no ion o a Λ(q) -se is, in some sense, lo al. Tha ollows om he Li lewo o d-Paley heo y. The nex p op osi ion is essen ially well-known, exep o he g ow h o he ons an , o whih we ha e ound no e e ene. Ao d- ingly, we oe a sho p o o . P op osi ion 1.1 Le Λ⊆[2,+∞[ . Then: (a) Le (Mn)n≥1 be a sequene o posi i e in ege s suh ha M1≤2 and Mn+1/Mn≥α > 1 . I Λ∩[Mn, Mn+1[ , n≥1 , has a uni o mly bounded Sidon ons an , hen Λ is Λ(q) o al l q≥2 ; mo e p eisely: k kq≤C(q, α)k k2 o e e y ∈ PΛ . (b) I Λ∩[2n,2n+1[ , n≥1 , has a uni o mly bounded Sidon ons an , Λ is Λ(q) o e e y q≥2 and, mo e p eisely: k kq≤Cq2k k2 o e e y ∈ PΛ and o some nume ial ons an C . P o o . (a) Se k=X Mk≤n<Mk+1 b (n)en and S =+∞ X k=1 | k|21/2. Sine Mk+1/Mk≥α > 1 and Λ⊆[M1,+∞[ , we ha e ([62℄, Chap. XV, Th. 2.1): k kq≤C0(q, α)kS kq. Now, using he 2-on exi y o he Lq -no m o q≥2 , we ob ain: kS kq≤+∞ X k=1 k kk2 q1/2. Bu k∈ PΛk , whe e Λk= Λ ∩[Mk, Mk+1[ has a uni o mly b ounded Sidon ons an . The e o e k kkq≤C1√qk k2 , whe e C1 is a nume ial ons an . The esul ollows. (b) We now make use o he lassial squa e un ion Sg =X k∈Z|gk|21/2, 5 whe e gk=X 2k≤n<2k+1 bg(n)en,i k≥0 and gk=X −2|k|+1<n≤−2|k|bg(n)eni k < 0. Fo his lassial squa e un ion, we ha e he ollowing sha p inequali y, due o J. Bou gain ([11℄, Th. 1): kSgkp≤C0(p−1)−3/2kgkp o 1 < p ≤2, whe e C0 is a nume ial ons an . We dedue by duali y ha : k kq≤C0q3/2kS kq o 2 ≤q < +∞. In a , by o hogonali y ( eall ha ∈ PΛ and ha Λ⊆[2,+∞[ ) and he CauhyShwa z inequali y, we ha e, o e e y g∈Lp wi h kgkp= 1 ( 1/p + 1/q = 1 ): |< , g > |= +∞ X k=1 < k, gk>=ZT +∞ X k=1 k(− )gk( )dm( ) ≤ZT S (− )Sg( )dm( ) ≤ kS kqkSgkp≤C0(p−1)−3/2kS kq ≤C0q3/2kS kq. This means ha he e we a e allowed o ake C0(q, 2) = C0q3/2 in pa (a) o he p o o . The es is unhanged, and we an also ake C(q, 2) = C1√qC0q3/2= Cq2 .  A se Λ⊆Z is alled a se o uni o m on e gene (in sho a UC-se ) i , o any ∈ CΛ , he symme i Fou ie sums SN( ) on e ge uni o mly o . I s ons an o uni o m on e gene U(Λ) is he smalles ons an K suh ha , o any ∈ CΛ : sup NkSN( )k∞≤Kk k∞. The ollowing a ian u ns ou o b e mo e a able ([56℄). Λ is alled a se o omple e uni o m on e gene (in sho a CUC-se ) i he ansla es ( Λ+a ) a e uni o mly UC o a∈Z , o equi alen ly, i he Fou ie sums SM,N ( ) on e ge uni o mly o as M, N go o +∞ , o e e y ∈ CΛ . The wo no ions u n ou o b e dis in ([20℄), bu lea ly oinide i Λ⊆N , whih will always be he ase in he sequel. The no ion o CUC-se is also a lo al one as he ollowing p op osi ion shows. P op osi ion 1.2 ([57℄, Th. 3) Le Λ⊆N∗ and ΛN= Λ ∩[N, 2N[ . (a) I U(ΛN) is bounded by K o N= 1,2,... , hen Λ is a CUC -se . (b) Le (Mn)n≥1 be a sequene o posi i e in ege s suh ha Mn+1/Mn≥2 . Then, i Λ∩[Mn, Mn+1[ a e quasi-independen o eah n , o mo e gene al ly i hey a e Sidon se s wi h uni o mly bounded Sidon ons an , hen Λ is a CUC -se . 6 Rema k. (b) is a use ul  i e ion o p o due se s ha a e CUC bu no Sidon; o ins ane, i Λ = S+∞ n=1{2n+ 2j;j= 0,...,n−1} , hen Λ∩[2n,2n+1[ is quasi-indep enden , whe eas Λ∩[1, N] has abou (log N)2 elemen s, and he e o e anno b e Sidon ( he mesh ondi ion o Sidon se s, see P op osi ion 1.6 b elow, is iola ed). The andom a iables whih we shall use will always be dened on some p obabili y spae (Ω,A,P) whih will play no explii ole, and he exp e a ion wi h esp e o P will always b e deno ed by E : E(X) = ZΩ X(ω)dP(ω). Reall he (mo e o less) lassial de ia ion inequali y (see [32℄,  6.3): Lemma 1.3 Le X1,...,XN be independen en e ed omplex andom a iables suh ha |Xk| ≤ 1 , k= 1,...,N . Le σ≥ N P k=1 E|Xk|2 . Then, one has, o e e y a≤σ : P(|X1+···+XN| ≥ a)≤4 exp(−a2/8σ). Le ( n)n b e a Be noulli sequene, i.e. a sequene o indep enden andom a iables suh ha : P( n= 1) = P( n=−1) = 1/2. Fo ∈ P , he spae o igonome i p olynomials, [[ ]] deno es he no m o in he Pisie 's spae C a.s. : [[ ]] = E X n nb (n)en ∞. See [25℄ and [47℄ o mo e in o ma ion ab ou his no m. Deni ion 1.4 A se Λ⊆Z is al led a p -Sidon se ( 1≤p < 2 ) i he e exis s a ons an K suh ha kb kp≤Kk k∞ o al l ∈ PΛ . I is said o be a p -Ride se i he e exis s a ons an K suh ha kb kp≤ K[[ ]] o al l ∈ PΛ . p -Ride se s we e implii ely in odued, wi h die en deni ion, in [18℄ (Th. 2.4), and in [23℄, p. 213, as lass Tp (see also [47℄, Th. 6.3). They we e explii ely dened and s udied in [51℄ and [52℄ unde he name  p -Sidon p esque sû s. We used almos su ely p -Sidon se  in he  s e sion o his pap e , bu , ollowing a sugges ion o J.-P. Kahane, we now use he e minology  p -Ride . Clea ly, e e y p -Sidon se is p -Ride . The on e se is ue o p= 1 : his is a ema kable esul due o D. Ride ([50℄), making le e use o D u y's on olu ion de ie (whih p o es ha he union o wo Sidon se s is Sidon [17℄). Whe he his on e se is s ill ue o 1< p < 2 is an op en p oblem. 7 Deni ion 1.5 We shal l say ha a ni e se B⊆Λ is M -pseudo-omplemen ed in Λ i he e exis s a measu e µ on T suh ha : |bµ| ≥ 1 on B;bµ= 0 on Λ B;kµk ≤ M . The ollowing p op osi ion gi es some neessa y, suien , o neessa y and suien ondi ions o a se Λ o b e p -Sidon o p -Ride . Pa (b) o his p op osi ion seems o b e new. P op osi ion 1.6 Le Λ⊆Z∗ and 1≤p < 2 . Se ε(p) = 2/p −1 . Then: (a) Λ is a p -Ride se i and only i he e exis s a ons an δ > 0 suh ha , o e e y ni e se A⊆Λ , he e exis s a quasi-independen subse B⊆A suh ha |B| ≥ δ|A|ε(p) . (b) Le q0>1 . I he e exis s a ons an δ > 0 suh ha , o e e y ni e se A⊆Λ , he e exis s a quasi-independen subse B⊆A suh ha |B| ≥ δ|A|1/q0 and i B an mo eo e be aken M -pseudo-omplemen ed in Λ , o some xed M , hen Λ is a q -Sidon se o e e y q > q0 . () I Λ is a p -Ride se , we ha e he ol lowing mesh ondi ion: |Λ∩[1, N]| ≤ C(log N)p/(2−p). P o o . We e e o [51℄ o he p o o o (a) and (). To p o e (b), le ∈ PΛ , x > 0 , and se A={|b |> } . Take B⊆A and µ as in Deni ion 1.4. Then B is a Sidon se wi h Sidon ons an ≤8 , and sine ∗µ=P n∈Bb (n)bµ(n)en , k k∞≥M−1  X Bb (n)bµ(n)en  ∞≥1 8MX B|b (n)||bµ(n)| ≥1 8MX B|b (n)| ≥ |B| 8M≥ δ|A|1/q0 8M· In o he wo ds, o some ons an C > 0 , one has: .|{|b |> }|1/q0≤Ck k∞, o e e y > 0, whih means ha he Lo en z no m o b in he Lo en z spae ℓq0,∞ is domina ed by k k∞ . Now, ℓq0,∞ is on inuously inje ed in ℓq o q > q0 (see o ins ane [38℄, I I p. 143), and his gi es he desi ed esul .  We deno e, as usual, by c0 he lassial spae o sequenes x= (xn)n≥0 ending o ze o a inni y, equipp ed wi h he no m kxk= supn|xn| . We say, in he usual amilia way, ha a Banah spae X on ains c0  i X has a losed subspae isomo phi o c0 . Ou no a ion o Banah spaes is lassial, as an b e ound in [16℄, [38℄ o [59℄ o ins ane. A subse Λ o Z is said o b e a Rosen hal se i e e y b ounded measu able un ion on T wi h spe um in Λ is almos e e ywhe e equal o a on inuous 8 un ion (in sho L∞ Λ=CΛ ). Λ is no Rosen hal i and only i L∞ Λ is no sepa able, so suh a se an b e hough as being a big se . E e y Sidon se is lea ly Rosen hal, bu H.P. Rosen hal ga e examples o non-Sidon se s whih a e Rosen hal ([53℄). We shall make use o he ollowing well known nega i e  i e ion (see [41℄,  3), whih ollows om he lassial heo em o C. Bessaga and A. Peªzy«ski ([38℄, I.2.e.8), saying ha a dual spae whih on ains c0 has o on ain also ℓ∞ . P op osi ion 1.7 I CΛ on ains c0 , hen Λ is no a Rosen hal se . Deni ion 1.8 Le Λ⊆N∗≡N {0} , and se ΛN= Λ ∩[1, N] and AN( ) = 1 |ΛN|X n∈ΛN en( ). We say ha Λ is: - e go di i (AN( ))N≥1 on e ges o a limi lΛ( )∈C o eah ∈T . - s ongly e go di i i is e godi and mo eo e he limi un ion lΛ denes an elemen o c0(T) : o e e y ε > 0 he se { ∈T;|lΛ( )|> ε } is ni e. - uni o mly dis ibu ed i i is (s ongly) e godi and, mo eo e , lΛ( ) = 0 o 6= 0 mod. 2π . The eason o his e minology is ha he e go di se s a e hose o whih an e go di heo em holds: (1/|ΛN|)Pn∈ΛNTn on e ges in he s ong op e a o op ology o e e y on a ion T o a Hilb e spae. Typially, he se o d h p e e p owe s, o he se o p ime numbe s a e s ongly e go di (ao ding o he esul o Vinog ado o i a ional mo d. 2π , and o he Di ihle 's a i hme i p og ession heo em o a ional mo d. 2π ). The hi d name omes om H. Weyl's lassial  i e ion o he equidis ibu ion o a eal sequene mo d. 2π . The ela ionship b e ween hese no ions omes om: Theo em 1.9 (F. Lus -Piqua d [42℄) Le Λ⊆[1,+∞[ be a se o posi i e in ege s. Then: (a) I Λ is s ongly e godi, CΛ on ains c0. (b) Mo e gene al ly, i Λ is s ongly e godi and D⊆Λ has a posi i e uppe densi y wi h espe o Λ , hen CD on ains c0 as wel l. He e p osi i e upp e densi y means ha : lim N→+∞|D∩[1, N]| |Λ∩[1, N]|>0. Pa (b) will b e use ul o us in he las heo em o Se ion 2. See [42℄ o he p o o o his heo em. The unde lying idea o (a) is ha i AN( )→lΛ( ) o e e y ∈T , lΛ denes an elemen o he bio hogonal C⊥⊥ Λ , and he ondi ion lΛ∈c0(T) implies ha i is he sum o a weakly un- ondi ionally Cauhy se ies o on inuous un ions. By using a p e u ba ion 9 Pu ν=δ0−VMn , whe e δ0 is he Di a p oin mass a 0 , and VMn he de la Vallée-Poussin ke nel o o de Mn . Conside he Riesz p o du R= Q k∈B (1 + Reek) , and se µ= 2ν∗R . We laim ha : kµk ≤ 8 ; bµ≥1 on B;bµ= 0 on Λ B. Indeed, kνk ≤ 4 and B is quasi-indep enden , so he Riesz p o du R e ies kRk=b R(0) = 1 . The e o e kµk ≤ 8 . Take l∈B . Then l > 2Mn and bν(l) = 1 . As b R(l)≥1/2 , we ha e bµ(l)≥1 . I A⊆Λ and |A|ε≤n1 , any single on B o A is quasi-indep enden , 1 - omplemen ed in Λ , and |B| ≥ n−1 1|A|ε . We ha e hus e ied he hyp o hesis o pa (b) o P op osi ion 1.6, and so Λ is q -Sidon o any q > 1/ε . In pa iula , i is p -Sidon, and his ends he p oo o Theo em 2.2.  Rema k 1. The p o o shows ha we an a ually ex a om A , o e e y α > 0 , a quasi-indep enden se B suh ha |B| ≥ δ|A|/(log |A|)2+α . Mo eo e , a sligh modia ion leads o se s e en lose o Sidon se s. P op osi ion 2.4 Le α > 1 and ϕα be he O liz un ion x7→ xlog(1+x)α . Then, he e exis s a se Λ as in Theo em 2.2 , and mo eo e suh ha b ∈ℓϕα,∞ o e e y ∈ CΛ . Reall ha ℓϕα,∞ is he weak O liz-Lo en z spae o sequenes (an)n suh ha sup n ϕ−1 α(n)a∗ n<+∞ , whe e (a∗ n)n is he non-in easing ea angemen o (|an|)n . The e o e, ano he way o ph ase he p op osi ion is, se ing an=b (n) : a∗ n≤Cαk k∞(log n)α/n o e e y ∈ CΛ. The p o o jus onsis s in hanging Mn . We ake Mn=en(log log n)2 , whe e [ ] s ands o he in ege pa . We s ill ha e P n PΩn(Mn)<+∞ , sine PΩn(Mn)≤2Cn nn (log Mnlog log Mn)n Mn≤exp −n(log log n)2/2 o n la ge enough. A guing as p e iously, we ge o e e y ni e subse A o Λ , a quasi-indep enden subse B o A suh ha |B| ≥ δ|A|/(log |A|)α , and suh ha CB is uni o mly pseudo-omplemen ed in CA . As in he p o o o P oposi ion 1.6, we ob ain |{|b |> }| ≤ C ϕαk k∞ , whih gi es he esul (a guing as in [34℄ o ins ane). We anno elimina e a loga i hmi a o , and eplae α > 1 by α > 0 b eause, due o Bou gain's  i e ion, we ha e o assume ha σn/log n go es o 16 inni y in o de ha CΛ on ains c0 . Howe e , o eah α > 0 , he e do exis non-Sidon se s Λ o whih b ∈ℓϕα when ∈ CΛ (as an b e seen om [5℄, p. 69). The se Λ is, in some sense, e y lose o b e Sidon, whe eas CΛ on ains c0 . Howe e , i anno be o o lose wi hou b eing Sidon beause i b ∈ℓ1,∞ , he Lo en z spae weak- ℓ1 , o e e y ∈ CΛ , hen Λ is Sidon. In a , his ondi ion implies an inequali y o he ype: |{|b | ≥ }| ≤ C k k∞(∗) o e e y ∈ CΛ . Le now A b e a ni e subse o Λ , and =P n∈A en and ω=P n∈A n(ω)en , whe e n , n≥1 a e he Rademahe un ions. Then, inequali y ( (∗) ) applied wi h = 1 gi es k ωk∞≥(1/C)|A| . In eg a ing in ω gi es [[ ]] ≥(1/C)|A| , om whih ollows, by a esul o G. Pisie ([48℄, Théo ème 2.3 ( i)), ha Λ is a Sidon se . Rema k 2. I one akes sele o s o mean δn suh ha nδn is b ounded, he o esp onding andom se Λ(ω) is almos su ely a Sidon se . This is a well-known esul o Y. Ka znelson and P. Mallia in ([28℄, o [27℄), and Lemma 2.1 gi es ano he p o o o his a . I sues o ake Mn=An , whe e A is a gi en in ege , la ge enough o ha e P+∞ n=1 PΩn(Mn)<1 . Then, wi h posi i e p obabili y Λ(ω)∩[Mn,+∞[ on ains no ela ion o leng h ≤n , whe eas |Λ(ω)∩[1, Mn]| ≤ Cn . Hene, o e e y ni e subse A o Λ(ω) , we an nd a quasi-indep enden subse B⊆A suh ha |B| ≥ δ|A| , o some xed δ=δ(ω) . I ollows om Pisie 's ha a e iza ion ([48℄, Th. 2.3 (i )) ha , wi h p osi i e p obabili y, and hene almos su ely by Kolmogo o 's 0−1 law, Λ(ω) is a Sidon se . As is now well-known, Sidon se s a e ha a e ized by a ious p op e ies (suessi ely weake ) o he Banah spae CΛ : Λ is a Sidon se i CΛ is isomo - phi o ℓ1 ([58℄), i CΛ has o yp e 2 ([31℄, Th. 3.1, [46℄), and i CΛ has a ni e o yp e ([13℄). This la e p op e y an b e exp essed by saying ha CΛ do es no on ain ℓn ∞ uni o mly. So, de e minis ially, one has he diho omy: (a) ei he Λ is a Sidon se , and so CΛ is isomo phi o ℓ1 ; (b) o CΛ on ains ℓn ∞ uni o mly. The p obabilis i diho omy is s onge : aking sele o s o mean δ1, δ2,... , wi h (δn)n de easing, one has: (a) ei he almos su ely Λ is a Sidon se (i nδn is b ounded); (b) o almos su ely CΛ on ains c0 (i nδn is no bounded), and Λ is e en uni o mly dis ibu ed. Y. Ka znelson ([27℄) al eady no ied suh a diho omy: he showed ha (unde a die en hoie o sele o s om ou s) ei he almos su ely Λ is a Sidon se , o almos su ely Λ is dense in he Boh g oup. Howe e , his is p e haps no a ue diho omy sine i is a well-known op en p oblem whe he he e an exis Sidon se s dense in he Boh g oup (see [15℄, ques ion 2, p. 14; 17 i is s a ed o he Boh g oup o R , bu also makes sense o he Boh g oup o Z ). The diho omy s a ed he e s eng hens Ka znelson's esul sine e e y uni- o mly dis ibu ed se is dense in he Boh g oup (see [6℄, Theo em 1); indeed, saying ha Λ = {λ1, λ2,...} is uni o mly dis ibu ed means ha he measu es µN= 1/N +∞ P n=1 δλn ( δλn is he e he Di a measu e a he poin λn ) on e ge weak-s a o he Haa measu e µ o he Boh g oup bZ ; bu hese measu es a e a ied by Λ , so he losed supp o o µ is on ained in he Boh losu e o Λ ; sine he Haa measu e is on inuous, we ge ha his losu e is he whole Boh g oup. Rema k 3. The andom se s Λ ha we ons u ha e an asymp o ial quasi- indep endene: Λ∩[Mn,+∞[ on ains no ela ion o leng h ≤n . This is eminis- en o he ollowing esul o J. Bou gain ([7℄): i Λ is a Sidon se and n∈N∗ , he e exis s ln=l(Λ, n) suh ha Λ an b e deomp osed in ln se s Λ1,...,Λln , eah o whih on ains no ela ion o leng h ≤n . We now in es iga e wha happ ens when we le p in ease away om 1 .We ge se e al die en esul s, and p= 4/3 seems o play a sp eial ole. We  s s a e wo e y simila esul s. Theo em 2.5 Fo e e y 1< p < 4/3 , he e exis s a se Λ o in ege s whih is: (1) uni o mly dis ibu ed (so Λ is dense in he Boh g oup, CΛ on ains c0 , and Λ is no a Rosen hal se ), and whih is: (2) Λ(q) o al l q < +∞ , a CUC -se , and mo eo e is: (a) p -Ride , bu no q -Ride o q < p (b) q -Sidon o al l q > p/(2 −p) . Theo em 2.6 Same as Theo em 2.5, exep ha , ins ead o (a) , Λ is: (a') q -Ride o e e y q > p , bu is no p -Ride . Rema k. A e his pap e was omple ed, P. Le è e and he hi d-named au ho p o ed ([36℄) ha e e y p -Ride se wi h p < 4/3 is a q -Sidon se , o all q > p/(2 −p) . A weake , unpublished, esul , due o J. Bou gain, is quo ed in [15℄, p. 41. Hene ondi ion (b) always ollows om ondi ion (a), and is no sp ei o he ons u ion. We do no know whe he his gap b e ween p and p/(2 −p) ollows only om ehnial easons. Fo p > 1 , whe he e e y p -Ride se is a ually p -Sidon is an op en ques ion. In Theo em 2.5, we ob ain se s whih a e p -Ride bu no q -Ride o q < p . We do no know i hese se s a e p -Sidon, so exa ly p -Sidon, in he e minology o R. Blei. He ons u ed suh se s using a ional p o du s ([3℄, [4℄). We may all he se s in Theo em 2.5  exa ly p -Ride se s . The se s app ea ing in Theo em 2.6 a e o a die en kind. We may all hem  exa ly p+ -Ride se s . Suh se s we e also ob ained in [3℄, Co ol. 1.7 d), whe e hey we e alled exa ly non- p -Sidon, and we e alled asymp o i p -Sidon in [5℄. 18 P o o . I is simila o ha o Theo em 2.2, so we shall be e y ske hy. Le α= 2(p−1)/(2 −p)∈] 0,1 [ . Fo Theo em 2.5, we use sele o s εk o mean δk=c(log k)α k(log log k)α+1 o k≥4. As in Lemma 2.3, we ha e, wi h Mn=nn , Pn≥1PΩn(Mn)<+∞ , and almos su ely C0nα+1 ≤ |ΛMn| ≤ C1nα+1 and |Λ′ n| ≤ C1nα o n la ge enough. Fo Theo em 2.6, we in ease he means δk sligh ly, eplaing hem by δk=c(log k)αlog log k k· Rema k. In o de o p o e ou heo ems, we used sele o s wi h a ious means. They a e smalle in Theo em 2.2 han in Theo em 2.5, o ins ane. We ema k ha sele o s (εk)k o mean δk wi h δk≤δ′ k may b e ahie ed as he p o du o wo indep enden sequenes o sele o s (ε′ k)k and (ε′′ k)k o mean δ′ k and δ′′ k= δk/δ′ k . I ollows ha , o example, he se s in Theo em 2.2 may b e ons u ed inside he esp e i e se s o Theo em 2.5. In Theo em 2.5, he p o o ha Λ was CUC o Λ(q) was based on he a ha |Λ′ n| ⊆ Λ∩[Mn,+∞[ is quasi-indep enden . Fo α≥1 ( i.e. p≥4/3 ), we no longe ha e |Λ′ n| ≤ n , and he e o e, a p io i , mus gi e up hese p op e ies. Howe e , we an use ano he ex a ion p oedu e. This p oedu e was  s in o dued by J. Bou gain ([8℄); la e , a lea s a emen was gi en in [52℄,  I I I.2. Sine his las e e ene is ha dly a ailable, we p e e o gi e a sel -on ained p o o . The o esp onding se Λ(ω) o in ege s ha we shall ob ain in his manne sa ises |Λ(ω)∩[2n,2n+1[| ∼ n∼log 2n , whih is he limi ing ondi ion o mesh (on a i hme i p og essions) o Sidon se s. This size is in some sense he la ges p ossible i we wan o ob ain a se Λ wi h blo ks ha ing a uni o mly b ounded Sidon ons an . Theo em 2.7 The e exis s a se Λ o in ege s whih is uni o mly dis ibu ed and on ains a subse E⊆N∗ whih is: (1) 4/3 -Ride , and no q -Ride o q < 4/3 ; a CUC -se ; a Λ(q) -se o al l q < +∞ (mo e p eisely, o al l q > 2 , we ha e: k kq≤Cq2k k2 o al l ∈ PE , whe e C > 0 is a nume ial ons an ), and ne e heless, (2) has posi i e uppe densi y in Λ , so, in pa iula , CE on ains c0 , and E is no a Rosen hal se . Le A b e a ni e subse o in ege s. Fo he p o o , i will be on enien o dene: ψA= sup p≥2 keAkp √p, whe e eA=X k∈A ek. We need he ollowing simple es ima e o ψA . 19 Lemma 2.8 Le I= [a+1, a+N] be an in e al o in ege s o leng h N , N≥3 . Then: ψI≤N √2 log N· P o o . Fo p≥2 , |eI|p≤Np−2|eI|2 , so R|eI|pdm ≤Np−2R|eI|2dm =Np−1 and keIkp/√p≤N1−1/p/√p . Op imizing gi es p= 2 log N ( ≥2 ), and he lemma.  This es ima e is essen ially op imal. Indeed, i is well-known ha ψI is uni o mly equi alen o θ=keIkΨ ( k kΨ b eing he no m asso ia ed o he O liz un ion Ψ(x) = ex2−1 ). Bu , o some ons an γ , |eI( )| ≥ γN o in an in e al J o leng h ≥γN−1 a ound 0 , so one has: 2≥ZJ exp |eI|2 θ2dm ≥γN−1exp γ2N2 θ2, whene θ≥γ−1N/plog 2γ−1N . We now use sele o s εk o mean δk=c n/2n o 2n≤k < 2n+1 , whe e c > 0 is a gi en ons an . Se In= [2n,2n+1[, n ≥2 ; δk=cn 2n i k∈In. No e ha (δk)k de eases, and δk is o he o m αk/k , whe e (αk)k go es o +∞ . I Λ = Λ(ω) is he o esp onding se o in ege s, i will b e on enien o se : Λn= Λ ∩In;σn=E|Λn|=X k∈In δk=cn . Fo his p o o , he alue o ψIn is somewha la ge, and equi es c b e su- ien ly small, say c≤1/576 . We p e e o ollow ano he ou e, whih ould b e use ul in o he on ex s, by ho osing also a andom se in In o whih he ψ ons an is small enough. We make he wo andom hoies a he same ime. Namely, we onside (ε′ n)n≥1 , a seond sequene o sele o s, indep enden o (εn)n≥1 , wi h xed mean τ , and se Λ′ n(ω) = {k∈Λn(ω) ; ε′ k(ω) = 1} . In sho : Λ′ n={k∈Λn;ε′ k= 1}; Λ′= +∞ [ n=1 Λ′ n. The ollowing lemma, whih is a sligh modia ion o Bou gain's ons u- ion in [8℄, is eally he hea o he p oo . Lemma 2.9 Almos su ely, o n la ge enough, one has: (1) (c/2) n≤ |Λn| ≤ (2c)n and (cτ/2) n≤ |Λ′ n| ≤ (2cτ)n (2) Λ′ n on ains a mos ela ions o leng h ≤ln , whe e ln= [144 c2τ2n] . 20 P o o o Lemma 2.9. We ha e al eady seen ha : P|Λn|−σn≥σn 2≤exp −σn 32 = exp −c n 32 , so, by he Bo el-Can elli lemma, |Λn| is almos su ely b e ween (c/2) n and (2c)n o n la ge enough; and his p o es he  s hal o (1). The seond hal holds o he same eason, sine Λ′ n o esp onds o sele o s εkε′ k wi h mean (cτ)n/2n o k∈In . The p o o o (2) is mo e elab o a e. Fix n , and onside he andom igonome i p olynomial: Fω= |In| X j=ln+1 X R⊆In |R|=jY k∈R εk(ω)ε′ k(ω)ek+e−k. Se : Nn(ω) = ZT Fω( )dm( ). Expanding Fω , we see ha : Fω( ) = |In| X j=ln+1 X R⊆In |R|=j X θk∈{−1,+1}RY k∈R εk(ω)ε′ k(ω)eθk k( ) = |In| X j=ln+1 X R⊆In |R|=jX θk∈{−1,+1}R ei Pk∈Rθkk. The on ibu ion o Nn(ω) o an exp onen ial o his sum is 0 i P k∈R θkk6= 0 , and is 1 i P k∈R θkk= 0 . The e o e, Nn(ω) is exa ly he numbe o ela ions o leng h > ln in Λ′ n . We laim ha Nn(ω) is almos su ely ze o o n la ge enough. To ha ee , we ma jo ize he exp e a ion J o Nn(ω) , using Fubini's heo em. Indeed, J=RTH( )dm( ) , whe e: H( ) = ZΩ Fω( )dP(ω) = |In| X j=ln+1 X R⊆In |R|=j δjY k∈R (ek+e−k). and δ=cτ n/2n . Hene: J= |In| X j=ln+1 X R⊆In |R|=j δjZTY k∈R(ek( ) + e−k( )dm( ). 21 A his s age, i is use ul o obse e ha : X R⊆In |R|=j ZTY k∈Rek( ) + e−k( )dm( ) ≤1 j!ZTX k∈In(ek( ) + e−k( )jdm( ). (3) Indeed, when we expand X k∈In(ek( ) + e−k( )j, eah e m Qk∈Rek( ) + e−k( ) app ea s j! imes, whe eas he o he e ms on he igh hand side o (3) a e p osi i e. I now ollows om (3) ha : J≤ |In| X j=ln+1 δj j!ZTX k∈In(ek( ) + e−k( )jdm( ) ≤ |In| X j=ln+1 δj j!2j  X k∈In ek   j j≤ |In| X j=ln+1 2jδj j!(ψInpj)j. Sine j!≥(j/e)j≥(j/3)j , his gi es J≤ +∞ X j=ln+1 6δψIn √jj≤ +∞ X j=ln+1 6δψIn √ln+ 1j. The e o e, J≤2−ln i 6δψIn √ln+ 1 ≤1 2 , i.e. i ln+ 1 ≥144(δψIn)2 . Bu , i ollows om Lemma 2.8 ha : ψIn≤2n p(2 log 2) n≤2n √n· The e o e 144(δψIn)2≤144cτ n 2n·2n √n2= 144 c2τ2n, and he hoie o ln jus  s o ob ain J≤2−ln . O ou se, we ha e assumed n la ge enough o ha e ln≥1 in ha p o o . Finally, Ma ko 's inequali y implies: X n≥2 P(Nn≥1) ≤X n≥2 ENn≤X n≥2 2−ln<+∞, 22 and by he Bo el-Can elli lemma, he in ege Nn is almos su ely ze o o n la ge enough, and ha ends he p o o o Lemma 2.9.  Now, using Bou gain's Theo em 1.10 and Lemma 2.9, one an nd Ω0⊆Ω wi h P(Ω0) = 1 suh ha o ω∈Ω0 , he e exis s n0=n0(ω) suh ha Λ = Λ(ω) and Λ′= Λ′(ω) sa is y: (4) Λ and Λ′ a e uni o mly dis ibu ed (5) (c/2) n≤ |Λn| ≤ (2c)n and (cτ/2) n≤ |Λ′ n| ≤ (2cτ)n o n > n0 (6) Λ′ n on ains a mos ela ions o leng h less han ≤ln= [144c2τ2n] o n > n0 . Λ′ n is no qui e quasi-indep enden , so we shall mo di y i sligh ly. We adjus one and o all τ , dep ending on c , suh ha 144c2τ2≤cτ/4 ( e.g. aking cτ = 1/576 ), so ha ln≤cτ n/4≤ |Λ′ n|/2 o n > n0 , in iew o (5). Sele hen in Λ′ n a ela ion R wi h supp o Sn o maximal a dinali y. Then |Sn| ≤ ln om (6), and En= Λ′ n Sn is quasi-indep enden . Mo eo e : |En|=|Λ′ n|−|Sn| ≥ |Λ′ n|−ln≥ |Λ′ n|/2 o n > n0 . Hene, i we se E=S n>n0 En , we ha e En=E∩In , and, mo eo e : (7) E has p osi i e upp e densi y in Λ (no e ha Λ′ has upp e densi y ≥τ/4 in Λ by (5)), (8) En is quasi-indep enden , (9) |En| ≥ (cτ/4) n , (10) I A⊆E is a ni e subse , hen A on ains a quasi-indep enden subse B wi h |B| ≥ (1/2) |A|1/2 . The las p op e y is p o ed in he ollowing way. Se Z={n;A∩En6=∅} and h=|Z| . We dis inguish wo ases. Case 1: he e exis s n∈Z suh ha |A∩En| ≥ |A|1/2 . Then, jus ake B=A∩En o ha e a quasi-indep enden se B suh ha |B| ≥ |A|1/2 . Case 2: |A∩En|<|A|1/2 o any n∈Z . Then h≥ |A|1/2 . W i e Z={n1<···< nh} , and pik an in ege mj∈ A∩Enj o eah j= 1,...,h . Then B={m1, m3,...}=: {µ1, µ2,...} is quasi- indep enden b eause we ha e µj+1/µj≥2 . Mo eo e |B| ≥ h/2≥(1/2) |A|1/2 . I is now easy o see ha E has he equi ed p ope ies. Indeed, i ollows om (4), (7), and om F. Lus -Piqua d's Theo em 1.9 ha E has a p osi i e upp e densi y in Λ . Tha i is CUC ollows om (8) and om P oposi ion 1.2. Tha i is Λ(q) o all q < +∞ ollows om (8) and om P op osi ion 1.1. 23 The a ha E is 4/3 -Ride ollows om (a) in P op osi ion 1.6. Indeed, i ε(p) = 2/p −1 , hen ε(4/3) = 1/2 . Finally, le N b e a la ge in ege , and n suh ha 2n≤N < 2n+1 . Then |E∩[1, N]| ≥ |En0+1|+···+|En−1| ≥ cτ 4(n0+ 1) + ···+ (n−1) ≥d0n2≥d1(log N)2 whe e d0, d1 a e p osi i e ons an s. I now E is a p -Ride se , we ha e he mesh ondi ion |E∩[1, N]|=O(log N)p/(2−p) . This equi es 2≤p/(2 −p) , ha is p≥4/3 . And his ends he p oo o Theo em 2.7.  Rema k. The hi d-named au ho p o ed he ollowing ([52℄, Lema 2.4) (whih is a ually implii ly al eady on ained in [48℄, Lemme 7.2, Théo ème 7.1, and Théo ème 2.3 (i )): ( ∗ ) Fo e e y ni e subse A⊆Z , he e exis s a quasi-indep enden subse B⊆A suh ha |B| ≥ δ(|A|/ψA)2 , whe e δ > 0 is a nume ial ons an . On he o he hand, G. Pisie ([47℄, Lemme 5.2) p o ed: E  X k ak kek Ψ≤CX k|ak|21/2 (1) whe e C is a nume ial ons an , ( k)k is he Rademahe sequene, and k kΨ is he O liz spae asso ia ed o Ψ(x) = ex2−1 . Taking ou sele o s εk wi h mean δk=c n/2n o k∈In , s anda d sym- me iza ion and en e ing a gumen s gi e: E  X k∈In εkek  Ψ≤C√n . (2) In o he e ms, we ha e, in iew o Lemma 2.9: E(ψΛn)≤C√n≤C′|Λn|1/2. (3) I we ould p o e a onen a ion inequali y, a ian o Lemma 1.3, hen his a ian and he Bo el-Can elli lemma would imply om (3) ha : Almos su ely ψΛn≤C′′|Λn|1/2 o n la ge enough. (4) We ould hen ombine ( ∗ ) and (4) di e ly o ob ain he ollowing al e na i e p o o o Theo em 2.7. Sele ω∈Ω suh ha Λ is s ongly e go di, wi h |Λn| ≥ c n/2 , and ψΛn≤C′′|Λn|1/2 ; ake hen a quasi-indep enden se En⊆Λn o size |En| ≥ δ|Λn| ψΛn2 ≥δC′′−2|Λn| ≥ δ′n; he se E=S n En hen has he equi ed p op e ies. 24 To end his se ion, we onside he ase p > 4/3 . We anno keep he p op e y o uni o m on e gene ( CUC ), no ha o b eing q -Sidon s a ed in Theo em 2.5. We do no know whe he his is only due o he me ho d. Bu b eing p -Ride wi h p > 4/3 migh b e a a he weak ondi ion (see [35℄ and [36℄). Theo em 2.10 Fo e e y 4/3≤p < 2 he e exis s a se Λ o in ege s whih is p -Ride , bu is no q -Ride o q < p and whih is Λ(q) o e e y q < +∞ , bu whih is uni o mly dis ibu ed (so in pa iula dense in he Boh g oup, and CΛ on ains c0 ). The p o o is essen ially he same as in Theo em 2.2, exep ha we ake sele o s εk o mean δk=c(log k)α k(log log k)α+1 o k≥1, whe e α= 2(p−1)/(2 −p)≥1 , and eplae Mn=nn by he smalles in ege ≥nβn , wi h β any numb e > α ( o ins ane β=α+ 1 ), whih we all again Mn . The es ima e: PΩn(Mn)≤2cn nn (log Mn)n(α+1) Mn s ill holds, and now gi es: PΩn(Mn)≤C′n(log n)n(α+1) n(β−α)n· Then easy ompu a ions show ha : ( ∗ ) Almos su ely |ΛMn| ∼ (nlog n)α+1 o n suien ly la ge; ( ∗∗ ) Almos su ely |Λ′ n| ∼ nα(log n)α+1 o n suien ly la ge. P op e y ( ∗ ) gua an ies ha Λ(ω) will s ill b e almos su ely p -Ride , and ( ∗∗ ) wi h he mesh ondi ion implies ha Λ is no q -Ride o q < p . The Λ(q) p op e y anno b e ob ained by he Li lewo o d-Paley me ho d, bu ollows om [43℄, Theo em 4.7.  3 La ge hin se s in p es ib ed se s o in ege s In his se ion, we s a om a p es ibed se Λ0={λ1< λ2< . . . < λN< ...} o p osi i e in ege s, and andomly ons u a hin se Λ inside Λ0 in he ollowing way. 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