Two random constructions inside lacunary sets

Tw o random constructions ins ide lacunary sets Stefan Neu wirth R ´ esum´ e Nous ´ etudions le rapp ort en tre la croissance d’une suite d’entiers et les propri´ et ´ es h armoniques et fonctio nnelles de la suite de ca ract` eres associ ´ ee. Nous mo ntrons en particulie r que toute suite p olynomiale, ai nsi que la suite des n ombres premiers, con tient un ensem ble Λ( p ) p our tout p qui n’est pas de Rosenthal. Abstract W e stu d y the relationship b etw een the grow th rate of an integer sequen ce and harmonic and functional prop erties of the corresp onding sequence of c haracters. W e sho w in particular that every p olynomial sequence contains a set that is Λ( p ) for all p b ut is not a R osenthal set. This holds also for the sequence of primes. 1 An in tro duction in F renc h 1.1 P osition du probl` eme Nous ´ etudions le rapp ort entre la croiss ance d’une suite { n k } = E ⊆ Z et deux de ses propr i´ et´ es harmoniques et fonctionnelles ´ even tuelles, i. e. toute fonction int ´ egr a ble sur le tore ` a sp ectre dans E est en fait p -int ´ eg rable p our tout p < ∞ : E est un ensem ble Λ( p ) po ur tout p ; toute fonction mesurable b orn´ ee sur le tore ` a s p ectr e dans E est contin ue ` a un ensemble de mesure nu lle pr` e s : E est un ensemble de Rosenthal. Nous sommes en mesur e de dr e s ser le tablea u s uiv ant selon la croissa nce po lynomiale: n k 4 k d po ur un d < ∞ , surp olynomiale: n k ≫ k d po ur tout d ≥ 1, sous-exp onentielle: log n k ≪ k , g´ eom´ etr ique: lim inf | n k +1 /n k | > 1. croissa nce po lynomiale surpo lynomiale e t so us-exp onentielle g´ eom´ etr ique E Λ( p ) ∀ p non presque toujours oui E Ro senthal presque jamais oui T able 1: C r oissance et propri´ et´ es harmo niq ues o u fonc tio nnelles. Li montre q u’effectivemen t il existe un ensemble Λ( p ) p our tout p qui n’est pas de Rosenthal. Nous traitons les deux questions s uiv antes. Question 1 .1 L e sch´ ema ci-dessus r este-t-il valable si on c onsid ` er e ` a la plac e de l’ensemble des sou s - ensembles E de Z l’ensemble des sous-ensembles E d’une suite ` a cr oissanc e p olynomiale ? Question 1 .2 Si E n ’est p as un ensemble de R osenthal, E c ontient-il un en semble ` a la fois Λ( p ) p our tout p et non Ro senthal ? 1 1.2 Constructions al´ eatoires ` a l’in t´ erieur de suites lacunaires Nous fournissons une preuv e nouv elle pour une construction al´ eato ir e d’ense mbles lacunaires par Yitzhak Katznelson qui appartient au folklore de l’analyse ha rmonique. Nous analyso ns et g´ en´ era lisons auss i la construction al´ eatoir e d’ensembles ´ equidistribu´ es par J e an B ourgain. Cela nous per met d’´ etablir le ta bleau 1.1 qui classe le s propri´ et´ es de Ros ent hal et Λ( p ) po ur tout p selon la cr oissance du sp ectre. Nous montrons a lo rs que la d´ emarche pro babiliste suivie pa r Katznelson et Bourgain p our construire c e s sous- ensembles de Z utilise seulement la crois sance “arithm´ etique” et l’´ e q uidistribution de la suite des entiers Z . En fait, ces sous-ensembles p euven t ˆ etre co nstruits ` a l’int ´ erieur de suites ´ equidistribu´ ees ` a croissa nce p olyno miale. En particulier , le ta ble a u 1.1 res te v ala ble p our l’ensemble des so us-ensembles E d’une suite p olynomia le , ainsi que de la suite des nombres premiers. Nous fournissons une r´ ep onse pa rtielle ` a la question 1.2 . Th ´ eor ` eme 1.3 Soit P une suite p olynomiale ou la suite des n ombr es pr emiers. Alors il existe une sous-suite E de P qui est Λ( p ) p our tou t p alors qu’el le ne forme p as u n ensemble de Ro senthal. 2 In tro duction The study of lacunar y sets in F ourier analy s is s till suffers from a severe la ck of examples, in par ticular for the purp ose o f distinguishing tw o pro pe rties. In order to bypass the individual complexity of int eger sets, one freq uently resorts to ra ndom constructions. In particular, Li [ 16 ] uses in his a r gumentation a construction due to K atznelson [ 13 ] to discr iminate the following tw o functiona l prop erties of c e r tain subsets E ⊆ Z : A Lebesg ue integrable function on the circle with F ourier frequencies in E is in fact p -in tegr able for all p < ∞ . This means that a ll spa ces L p E ( T ) co incide for p < ∞ , i. e. E is a Λ( p ) se t for all p in Rudin’s terminolo gy . No sequence of p olynomial growth ha s this prop erty [ 24 , T h. 3 .5 ]. By Theor em 5.7 , a lmost e very sequence of a given s uper p olynomial order of growth is Λ( p ) for all p . A bo unded measurable function o n the c ir cle with F ourier frequencies in E is in fact contin uous up to a set o f measure 0. This mea ns that L ∞ E ( T ) and C E ( T ) coincide: E is a Rosenthal set. Every seq uence of exp one ntial growth is a Sidon set and therefore has this prop erty . By Bour gain’s Theo rem 3.5 , almost every sequence of a given sub exp onential order o f gr owth fails the Rosenthal prop er t y . A Ros e n thal set may co nt ain ar bitrarily large interv als [ 23 ] und thus fail the Λ( p ) prope r ty . This shows that these tw o pr o p erties cannot be characteriz ed by some order of growth, wher eas the r a ndom metho d is so imprecis e that it ignor es a r ange of exceptiona l s ets. On the other hand, Li shows that some set E is Λ( p ) for a ll p and fails the Ro senthal prop er t y: his construction witnesses for the quantitativ e overlap betw een sup erp oly no mial and sub exp o nential order of growth. F rom a Banach space po int of v iew, Li’s set E is such tha t C E ( T ) c o ntains c 0 while L 1 E ( T ) do es no t contain ℓ 1 . W e c ome back to Li [ 16 ] for tw o reasons : in the first place, we have b een unable to lo cate a published pro o f of Katznels o n’s statement. W e provide one for a strong er statemen t in Section 5 . In the second place , we wan t to precise and supple the random construction in the following sense: can one distinguish the Λ( p ) prop erty a nd the Rosenthal prop erty among subsets of a c e rtain g iven s e t ? That s ort o f questions has been inv estigated by Bourg a in in [ 4 ]. W e give the following answer (see Th. 3.8 ): Main Theorem Consider a p olynomial se quenc e of inte gers, or the se quen c e of primes. Then some subse quenc e of it is Λ( p ) for al l p and at the same time fails the Ro senthal pr op erty. This is a sp ecial case of the mor e gener al question: do es every s e t that fails the Rosenthal pr op erty contain a subset that is Λ( p ) fo r all p and still fails the Rosenthal prop erty ? W e should emphasize at this p oint that neither of these notions has an arithmetic description. In fact, the family o f Rose nthal sets is coanalytic non Borel [ 9 ] and a ny description w ould b e at lea st as complex as their definition. This is why we study instead the following t wo prop erties for certain subsets E ⊆ Z . An y integer n has at most one re pr esentation as the sum of s elements of E . This implies that E is Λ(2 s ) b y [ 24 , Th. 4.5(b)]. 2 E is equidistributed in Hermann W eyl’s sense: sav e for t ≡ 0 mo d 2 π , the s uc c e ssive means of { e i nt } n ∈ E tend to 0, whic h is the mean of e i t ov er [0 , 2 π [. This implies that E is not a Rosenthal set by [ 18 , Lemma 4 ]. Our random constructio n gives no hint fo r explicit pro cedures to build such integer sets . The question whether some “natural” set of integers is Λ( p ) for all p a nd fails the Ros e nthal pr op erty r emains op e n. Let us describ e the pap er br iefly . Section 3 int ro duce s the inquired notions and gives a s urvey o f for mer and new r esults. As the r ig ht fra mework for this study a pp ea rs to co nsist in the sequences of p olynomial growth, we give them a precise mea ning in Section 4 , and show that they are nicely distr ibuted among the interv als of the pa rtition of Z defined by {± 2 k ! } . Sectio n 5 establishes a n optimal criterio n fo r the generic subset of a set with poly nomial growth to b e Λ( p ) for a ll p . Section 6 comes ba ck to Bo ur gain’s pro of in [ 3 , Pr op. 8.2(i)]: we simplify and strengthen it in o r der to inv estigate the generic subset o f an equidistributed set. Notation T = { t ∈ C : | t | = 1 } is the unit circle endow ed with its Haar measure dm and Z its dual group of integers: for each n ∈ Z , let e n ( t ) = t n . The car dina l of E = { n k } ⊆ Z is written E . W e denote by c 0 ( T ) the spa ce o f functions on T which are arbitrar ily small outside finite sets; such functions necessarily hav e co unt able supp ort. F or a space o f integrable functions on T and E ⊆ Z , X E denotes the space of functions with F ourier sp ectrum in E : X E =  f ∈ X : b f ( n ) = R e − n f dm = 0 if n / ∈ E  . W e shall stick to Hardy’s notation: u n 4 v n ( vs. u n ≪ v n ) if u n /v n is b ounded ( vs. v anishes) a t infinity . Ac kno wledgment I would like to thank Daniel Li for several helpful discussions. 3 Equidistributed and Λ( p ) sets Definition 3. 1 L et E = { n k } k ≥ 1 ⊆ Z or der e d by incr e asing absolute value | n k | . ( i ) [ 24 , Def. 1.5] L et p > 0 . E is a Λ( p ) set if, for s ome — or e quivalently for any — 0 < r < p , L p E ( T ) and L r E ( T ) c oincide: ∃ C r ∀ f ∈ L p E ( T ) k f k r ≤ k f k p ≤ C r k f k r . ( ii ) [ 26 , § 7] E is e quidistribute d if for e ach t ∈ T \ { 1 } the suc c essive me ans f k ( t ) = 1 k k X j =1 e n j ( t ) − − − → k →∞ 0 . (1) Thus E is e quidistribute d if and only if the se quenc e of char acters in E c onver ges to 1 { 1 } for the Ces` ar o summing metho d. If f k tends p ointwise to f ∈ c 0 ( T ) , then E is we akly e quidistribute d. If E is weakly equidistributed, then f defines a n element o f C E ( T ) ⊥⊥ . By Lust- Piquard’s [ 18 , Lemma 4], C E ( T ) then cont ains a copy o f c 0 and E ca nnot b e a Rosenthal set. F or example, Z and N are equidis tr ibuted. Arithmetic sequences are weakly equidistributed: there is a finite set on which f k 9 0 . Polynomial sequences of integers ([ 26 , Th. 9] and [ 25 , Lemma 2.4], see [ 19 , Ex. 2 ]) and the sequence of prime num b ers (Vinog radov’s theorem [ 5 ], see [ 19 , Ex. 1 ]) ar e weakly equidistributed: f k ( t ) may no t conv erge to 0 fo r r ational t . There are nevertheless seq ue nc e s of b ounded pace that ar e not weakly eq uidis tr ibuted [ 8 , Th. 11]. Sidon sets a re Λ( p ) for all p [ 24 , Th. 3.1 ], but not weakly equidistributed since they are Rosenthal s ets. Example 3.2 Co ns ider the geometric sequence E = { 3 k } k ≥ 1 and the co rresp onding sequence o f suc- cessive means f k . By [ 8 , Th. 1 4], the f k do not conv erge to 0 on a null se t of Haus do rff dimension 1. Consider f j k = k − j X 1 ≤ k 1 ,...,k j ≤ k e 3 k 1 + ··· +3 k j = k − j  j ! X 1 ≤ k 1 < ··· p 2 k − 1 ( k ≥ 1) . L et E = N in Constru ction 3.4 . Ther e is a choic e of ( ℓ k ) with ℓ k ≫ lo g p k such that for δ n = ℓ k / I k ( n ∈ I k ) , E ′ is Λ( p ) for al l p almost sur ely. Li suggests to apply the conten t of Pro p o s ition 3.7 with p k = 2 k and ℓ k = k : then δ n ≫ n − 1 and Theorem 3.3 derives fr o m Theorem 3 .5 . W e shall genera lize K atznelson’s and Li’s r e s ults with a new pr o of that p ermits to construc t E ′ inside of sets E with p olynomia l growth (see Def. 4.1 ) and yields an optimal cr iterion o n ℓ k . W e sha ll subsequently generalize Bourgains’s Theorem 3.5 to o bta in the Main Theor em via Theorem 3 .8 L et E b e e quidistribute d ( vs. we akly ) and with p olynomial gr owth. Then t her e is a subset E ′ ⊆ E e quidistribute d ( vs. we akly ) and at the same time Λ( p ) for al l p . A precise and quant itative statement of this is Theorem 6.5 . 4 Sets with p olynomial gro wth W e star t with the definition and firs t prop erty of such sets. 4 Definition 4. 1 L et E = { n k } k ≥ 1 ⊆ Z b e an infinite set or der e d by incr e asing absolute value and E [ t ] = E ∩ [ − t, t ] its distribution function. ( i ) E has p olynomial gr owth if n k 4 k d for some 1 ≤ d < ∞ . This amounts to E [ t ] < t ε for ε = d − 1 . ( ii ) E has r e gular p olynomial gr owth if ther e is a c > 1 such that | n ⌈ ck ⌉ | ≤ 2 | n k | for lar ge k . This amounts to E [2 t ] ≥ cE [ t ] for lar ge t . Pr o of. ( i ) If | n k | ≤ C k d for large k and C k d ≤ t < C ( k + 1 ) d , then E [ t ] ≥ k > ( t/ C ) ε − 1. C o nv ersely , if E [ t ] ≥ ct ε for large t and c ( t − 1 ) ε < k ≤ ct ε , then | n k | ≤ t < ( k /c ) d + 1. ( ii ) If | n ⌈ ck ⌉ | ≤ 2 | n k | for large k and k is maximal with | n k | ≤ t , then E [2 t ] ≥ E [2 | n k | ] ≥ ck = cE [ t ]. Conv ersely , if E [2 t ] ≥ cE [ t ] for lar ge t , then E [ | n k | ] ∈ { k , k + 1 } and E [2 | n k | ] ≥ ck . Thus | n ⌈ ck ⌉ | ≤ 2 | n k | . In particular , po lynomial sequences have reg ular po lynomial growth. By the Prime Number Theorem, the sequence of primes a lso ha s. Proper t y ( ii ) implies pro per ty ( i ): if E [2 t ] ≥ cE [ t ] for large t , then E [ t ] < t log 2 c . The converse howev er is false as shows F = S ]2 2 2 k , 2 2 2 k +1 ], for which F [ t ] < t 1 / 4 while F [2 t ] = F [ t ] infinitely often. Let us relate Definition 4.1 with cer tain partitio ns o f Z . Regular gr owth means in fact that E is regular ly distributed on the annu lar dyadic partition of Z P =  [ − p 0 , p 0 ] , I k = [ − p k , − p k − 1 [ ∪ ] p k − 1 , p k ]  k ≥ 1 where p k = 2 k (2) and F shows that there are sets with po lynomial g rowth which are no t regular ly distributed on the partition defined by p k = 2 2 k . How ever, the interv als of the gro ss par tition P =  [ − p 0 , p 0 ] , I k = [ − p k , − p k − 1 [ ∪ ] p k − 1 , p k ]  where log p k ≫ log p k − 1 (3) grow with a sp eed that for ces reg ularity . Put p k = 2 k ! for a simple explicit exa mple. W e have pr e cisely Prop ositi o n 4.2 L et E ⊆ Z , P = { I k } a p artition of Z and E k = E ∩ I k . Then log E k < log I k in the two fol lowing c ases: ( i ) E has r e gular p olynomial gr owth and P is p artition ( 2 ) ; ( ii ) E has p olynomial gr owth and P is p artition ( 3 ) . Pr o of. ( i ) Cho os e K and c > 1 suc h that E [2 k ] ≥ cE [2 k − 1 ] for k ≥ K . Then E [2 k ] < c k . Thus E k = E [2 k ] − E [2 k − 1 ] ≥ (1 − c − 1 ) E [2 k ] < c k = 2 k log 2 c . ( ii ) In this case p ε k ≫ p k − 1 for each po sitive ε . Now ther e is ε > 0 s uch that E k = E [ p k ] − E [ p k − 1 ] < p ε k < I k ε . 5 Sets that are Λ( p ) for all p In this section, we establish a n improv ement (Th. 5.7 ) of Katznelson’s statement [ 13 , § 2]. W e first recall several known definitions and results. Λ( p ) sets hav e a practical desc ription in terms o f unconditionality . W e shall a lso use a combinatorial prop erty that is more elementary than [ 24 , 1.6(b)]: to this end, write Z m s for the following set of arithmetic relations. Z m s =  ζ ∈ Z ∗ m : ζ 1 + · · · + ζ m = 0 and | ζ 1 | + · · · + | ζ m | ≤ 2 s  . Note that Z 1 s and Z m s ( m > 2 s ) ar e empty , and that every ζ ∈ Z 2 s is of form ζ 1 · (1 , − 1): this is the ident ity r elation. Definition 5. 1 L et 1 ≤ p < ∞ , s ≥ 1 inte ger and E ⊆ Z . ( i ) [ 12 ] E is an un c onditional b asic se quenc e in L p ( T ) if sup ±     X n ∈ E ± a n e n     p ≤ C     X n ∈ E a n e n     p . for some C . If C = 1 works, E is a 1 -u nc onditional b asic se quenc e in L p ( T ) . ( ii ) E is s -indep endent if P m 1 ζ i q i 6 = 0 for al l 3 ≤ m ≤ 2 s , ζ ∈ Z m s and distinct q 1 , . . . , q m ∈ E . 5 Prop ositi o n 5.2 L et 1 ≤ p 6 = 2 < ∞ , s ≥ 1 inte ger and E ⊆ Z . ( i ) [ 24 , pr o of of Th. 3.1] E is a Λ (max( p, 2)) set if and only if E is an unc onditional b asic se quenc e in L p ( T ) . ( ii ) [ 22 , Pr op. 2.5, R em. (1)] E is a 1 -unc onditional b asic se quenc e in L 2 s ( T ) if and only if E is s -indep endent. W e need to introduce a s econd clas sical notio n of unconditionality that rests on the Littlewoo d–Paley theory . Definition 5. 3 ([ 11 ]) L et P = { I k } b e a p artition of Z in fin ite intervals. P is a Littlewo o d–Paley p artition if for e ach 1 < p < ∞ ther e is a c onstant C p such t hat ∀ f ∈ L p ( T ) sup ±    X ± f k    p ≤ C p k f k p with b f k =  b f on 0 off I k . (4) By Khinchin’s inequality , this means exactly that ∀ f ∈ L p ( T ) k f k p ≈     X | f k | 2  1 / 2    p . In par ticular, the dyadic partition ( 2 ) and the gross par tition ( 3 ) are Littlewoo d– Paley [ 17 ]. B y Pro p o- sition 5.2 and ( 4 ), w e obtain Prop ositi o n 5.4 L et { I k } b e a Littlewo o d–Paley p artition and E k ⊆ I k . If E k is s -indep endent for e ach k , then E = S E k is an unc onditional b asic se quenc e in L 2 s ( T ) and thus a Λ(2 s ) set. W e ge ne r alize now K atznelson’s Prop o s ition 3.7 . Lemma 5. 5 L et s ≥ 2 inte ger, E ⊆ Z finite and 0 ≤ ℓ ≤ E . Put δ n = ℓ/ E in Construction 3.4 , so that al l sele ctors ξ n have same distribution. Then t her e is a c onstant C ( s ) that dep ends only on s such that P [ E ′ is s -dep endent ] ≤ C ( s ) ℓ 2 s E . Pr o of. W e wish to co mpute the probability that there are 3 ≤ m ≤ 2 s , ζ ∈ Z m s and distinct q 1 , . . . , q m ∈ E ′ with P ζ i q i = 0 . As the num b er C ( s ) of arithmetic relations ζ ∈ Z m s (3 ≤ m ≤ 2 s ) is finite and depe nds o n s o nly , it suffices to co mpute, for fixed m and ζ ∈ Z m s P h ∃ q 1 , . . . , q m ∈ E ′ distinct : X ζ i q i = 0 i = P " ∃ q 1 , . . . , q m − 1 ∈ E ′ distinct : − ζ − 1 m m − 1 X i =1 ζ i q i ∈ E ′ \ { q 1 , . . . , q m − 1 } # = P    [ q 1 ,...,q m − 1 ∈ E ′ distinct h − ζ − 1 m m − 1 X i =1 ζ i q i ∈ E ′ \ { q 1 , . . . , q m − 1 } i    = P    [ q 1 ,...,q m − 1 ∈ E distinct h q m = − ζ − 1 m m − 1 X i =1 ζ i q i ∈ E \ { q i } m − 1 i =1 & ξ q 1 = · · · = ξ q m = 1 i    . The union in the line ab ov e runs o ver E ! ( E − m + 1)! ≤ E m − 1 ( m − 1 )-tuples. F urther, the even t in the inner brack ets implies that m o ut of E selecto r s ξ n hav e v alue 1: its pro bability is b ounded by ( ℓ/ E ) m . Thus P [ E ′ is s -dep endent ] ≤ C ( s ) max 3 ≤ m ≤ 2 s E m − 1 ℓ m E m ≤ C ( s ) ℓ 2 s E . The random metho d we shall use is the following random construction. 6 Construction 5 .6 L et E ⊆ Z . L et { I k } b e a Littlewo o d–Paley p artition and E k = E ∩ I k . L et ( ℓ k ) k ≥ 1 with 0 ≤ ℓ k ≤ E k and put P [ ξ n = 1 ] = δ n = ℓ k / E k ( n ∈ E k ) in Constru ction 3.4 . Put E ′ k = E ′ ∩ I k . Theorem 5 .7 L et E ⊆ Z have p olynomial ( vs. r e gular ) gr owth and { I k } b e the gr oss ( 3 ) ( vs. dyadic ( 2 )) Littlewo o d–Paley p artition. Do Construction 5.6 . The fol lowing assertions ar e e quivalent. ( i ) lo g ℓ k ≪ log I k , i. e. log ℓ k ≪ log p k ( vs. log ℓ k ≪ k ) ; ( ii ) E ′ is almost sur ely a Λ( p ) set for al l p . Pr o of. Note that by Pr o p osition 4 .2 , ther e is a p ositive α such that E k > I k α for large k . ( i ) ⇒ ( ii ) Let s ≥ 2 be a n a rbitrary in teger . By Prop osition 5.5 , ∞ X k =1 P [ E ′ k is s -dep endent] ≤ C ( s ) ∞ X k =1 ℓ 2 s k E k . F or each η > 0 , ℓ k ≤ I k η for large k . Cho ose η < α/ 2 s . Then ℓ 2 s k / E k ≤ I k 2 sη − α for large k , and the series above conv erges s ince I k < 2 k . By the Bor el–Cantelli lemma, E ′ k is almos t surely s -indep endent for larg e k . By Prop ositio n 5.4 , E ′ is almost s ur ely the union o f a finite set and a Λ(2 s ) set. By [ 2 4 , Th. 4.4(a)], E ′ itself is almost surely a Λ(2 s ) set. ( ii ) ⇒ ( i ) If E ′ is a Λ(2 s ) set, then by [ 24 , Th. 3.5] or simply by [ 4 , (1 .12)], there is a co ns tant C s such tha t E ′ k < C s I k 1 /s . As E ′ k ∼ ℓ k almost surely by the Law of Large Numbers (cf. the following Lemma 6.1 ), one ha s log ℓ k ≪ log I k . Remark 5.8 As one may easily co nstruct sets that gr ow a s slowly as one wis hes a nd nev ertheless contain arbitrarily la rge interv als (se e als o [ 24 , Th. 3.8] for an o ptimal statement), one cannot r emov e the adverb “almos t surely” in Theo r em 5.7 ( ii ). Remark 5.9 The right formulation o f Ka tznelson’s Prop ositio n 3.7 thus turns out to be the following. Let E = N and I k = ] p k − 1 , p k ] with p k > cp k − 1 for some c > 1 in Constr uction 5.6 . Then E ′ is almost surely a Λ ( p ) set for a ll p if and only if log ℓ k ≪ log p k . Remark 5.10 Theore m 5.7 shows that there are sets that are Λ ( p ) for all p of any given sup erp olynomial order of growth. This is optimal since s e ts with distribution E [ t ] < t ε fail the Λ( p ) prop erty for p > 2 /ε by [ 24 , Th. 3.5]. Such s e ts may also b e c o nstructed inductively by co m binator ial means: see [ 10 , § II, (3.52)]. 6 Equidistributed sets In this se ction, we shall finally sta te and prove our pr incipal r esult. T o this end, we shall first generalize Bourga in’s Theorem 3.5 in or der to get Theor em 6.4 . The following lemma is Bernstein’s distribution ineq uality [ 2 ] and dates back to 1924. Lemma 6. 1 L et X 1 , . . . , X n b e c omplex indep endent r andom variables such that | X i | ≤ 1 and E X i = 0 and E | X 1 | 2 + · · · + E | X n | 2 ≤ σ. (5) Then, for al l p ositive a , P [ | X 1 + · · · + X n | ≥ a ] < 4 exp ( − a 2 / 4( σ + a )) . (6) Pr o of. Co nsider first the cas e o f rea l r andom v aria bles. By [ 1 , (8b)], P [ X 1 + · · · + X n ≥ a ] < exp( a − ( σ + a ) log(1 + a/σ )); as log(1 − u ) ≤ − u − u 2 / 2 for 0 ≤ u < 1, P [ X 1 + · · · + X n ≥ a ] < exp( − a 2 / 2( σ + a )) . 7 One gets ( 6 ) since for complex z | z | ≥ a = ⇒ max( ℜ z , −ℜ z , ℑ z , −ℑ z ) ≥ a/ √ 2 . The next lemma corr esp onds to [ 3 , Lemma 8.8 ] and is the cruc ia l s tep in the estimation of the succ essive means of { e i nt } n ∈ E ′ . Note that its h yp othesis is not on the individual δ n , but on their successive s ums σ k : this is needed in order to cop e with the irregula rity of E . Lemma 6. 2 L et E = { n k } ⊆ Z b e or der e d by incr e asing absolute value. Do Construction 3.4 and put σ k = δ n 1 + · · · + δ n k . If σ k ≫ log | n k | , then almost sur ely ψ ( k ) =     1 E ′ ∩ { n 1 , . . . , n k } X e n E ′ ∩{ n 1 ,...,n k } − 1 σ k k X j =1 δ n j e n j     ∞ − − − → k →∞ 0 . (7) Pr o of. Note that X e n E ′ ∩{ n 1 ,...,n k } = k X j =1 ξ n j e n j , E ′ ∩ { n 1 , . . . , n k } = k X j =1 ξ n j . Cent er the ξ n by putting f = P k j =1 ( ξ n j − δ n j ) e n j . Then ψ ( k ) ≤     E ′ ∩ { n 1 , . . . , n k } − 1 − σ − 1 k  k X j =1 ξ n j e n j    ∞ + k σ − 1 k f k ∞ ≤ σ − 1 k     δ n 1 + · · · + δ n k ξ n 1 + · · · + ξ n k − 1     k X j =1 ξ n j + σ − 1 k k f k ∞ ≤ 2 σ − 1 k k f k ∞ . Let R = { t ∈ T : t 4 | n k | = 1 } and u ∈ T s uch that | f ( u ) | = k f k ∞ . Let t ∈ R b e at minimal distance of u : then | t − u | ≤ π / 4 | n k | . By Bernstein’s theorem, k f k ∞ − | f ( t ) | ≤ | f ( u ) − f ( t ) | ≤ | t − u | k f ′ k ∞ ≤ 4 5 k f k ∞ ; k f k ∞ ≤ 5 sup t ∈ R | f ( t ) | . (F o r an optimal b ound, cf. [ 21 , § I.4, Lemma 8 ].) F or each t ∈ R , the rando m v ariables X j = ( ξ n j − δ n j ) e n j ( t ) satisfy ( 5 ), so that P [ | f ( t ) | ≥ a ] < 4 exp( − a 2 / 4( σ k + a )) . As R = 4 | n k | , P [ k f k ∞ ≥ 5 a ] ≤ P  sup t ∈ R | f ( t ) | ≥ a  < 4 | n k | · 4 exp( − a 2 / 4( σ k + a )) . Put a k = (1 2 σ k log | n k | ) 1 / 2 . Then a k ≪ σ k : therefor e P [ k f k ∞ ≥ 5 a k ] 4 | n k | − 2 and by the Borel–Cantelli lemma, σ − 1 k k f k ∞ 4 a k /σ k − − − → 0 almost s urely . Remark 6.3 The h yp othesis in Lemma 6.2 contains implicitly a re s triction on the lacuna r ity of E . If σ k ≫ log | n k | , then necess a rily log | n k | ≪ k a nd E [ t ] ≫ lo g t . In pa rticular, E cannot be a Sidon se t by [ 24 , Cor. of Th. 3.6]. W e now sta te and pr ove the equidistr ibuted count erpa rt of Theor e m 5.7 . 8 Theorem 6 .4 L et E = { n k } ⊆ Z b e e quidistribute d ( vs. we akly ) , and or der e d by incr e asing absolute value. Do Construction 3.4 and supp ose that δ n j de cr e ases with j . Put σ k = δ n 1 + · · · + δ n k . If σ k ≫ log | n k | , then E ′ is almost sur ely e quidistribute d ( vs. we akly ) . This is in p articular t he c ase if ( a ) δ n k ≫ ( | n k | − | n k − 1 | ) / | n k − 1 | ; ( b ) E has p olynomial gr owth and δ n k ≫ k − 1 . Pr o of. Lemma 6.2 shows that almost sur ely ( 7 ) holds. It remains to show that lim 1 σ k k X j =1 δ n j e n j = lim 1 k k X j =1 e n j , i. e. that the matrix summing method ( a k,j ) given by a k,j = ( δ n j /σ k if j ≤ k 0 if not is r egular a nd stro nger than the C e s` a ro metho d C 1 by ar ithmetic means. This is the case b eca use a k,j ≥ 0 , P j a k,j = 1 and (cf. [ 27 , § 52, Th. I]) ∀ k X j j | a k,j − a k,j +1 | = X j j ( a k,j − a k,j +1 ) = 1 < ∞ since a k,j decreases with j for each k . ( a ) In this ca se δ n k ≫ log | n k | − log | n k − 1 | a nd th us σ k ≫ log | n k | . ( b ) In this c ase, σ k ≫ log k < log | n k | . In conclusion, w e obtain, by combining Theorems 5.7 and 6.4 , our principal result. Theorem 6 .5 L et E ⊆ Z b e e quidistribute d ( vs. we akly ) and do Constr u ction 5.6 . Then E ′ is almost sur ely Λ( p ) for al l p and at the same time e quidistribute d ( vs. we akly ) in the two fol lowing c ases: ( i ) E is a set of r e gular p olynomial gr owth, { I j } is the dyadic Littlewo o d–Paley p artition ( 2 ) , 1 ≪ log ℓ j ≪ j and ℓ j / E j de cr e ases eventual ly. ( ii ) E is a set of p olynomial gr owth, { I j } is the gr oss Littlewo o d–Paley p artition ( 3 ) , ℓ j / E j de cr e ases eventual ly and ℓ j ≫ log p j +1 while lo g ℓ j ≪ log p j . T his is the c ase if we put p j = 2 j ! and ℓ j = min(( j + 2 )! , E j ) . Pr o of. In ea ch case log ℓ j ≪ lo g I j . Let us show that the h yp othesis o f Theorem 6.4 is v erified. If n k ∈ E j ⊆ I j , then | n k | ≤ p j and σ k ≥ j − 1 X i =1 X n ∈ E i δ n = ℓ 1 + · · · + ℓ j − 1 and in each case ℓ j − 1 ≫ log p j − log p j − 1 . 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