Large fluctuations of sums of a random multiplicative function

2 log 2 − 1 b y Soundarara jan and Xu [42], entering a regime of H where naiv ely the fourth moment of the sum blows up. How ever, 1 An in teger is called y -smo oth if all of the prime factors dividing it are at most y . 4 BESFOR T SHALA this is bypassed b y throwing out 0% of the in tegers in the short in terv al with at ypically man y prime factors, resulting in a con trolled fourth moment. W e pro ve an almost sure low er b ound in the same regime. Theorem B. L et f b e a R ademacher or Steinhaus r andom multiplic ative function and supp ose that H = H ( N ) is a smo oth, incr e asing, and c onc ave function, such that N 11 15 ≤ H ( N ) ≤ N / (log N ) c for some c onstant c > 2 log 2 − 1 . Then almost sur ely ther e exists a se quenc e N k → ∞ with c orr esp onding H k = H ( N k ) such that       X N k − H k 2 log 2 − 1, satisfy these conditions (for sufficiently large N ). W e ha v e not attempted to optimize the exp onen t 11 15 – it is likely p ossible to bring it down to 3 5 . How ever, the lo wer b ound on H should not b e essential: for very small H ≪ log N the large fluctuations are captured b y almost sure long runs of ones, of length ≈ log N (see the w ork of Erd˝ os and R ´ enyi [13]), whereas if log N ≪ H ≪ N 11 15 , our metho d should w ork (with significan t simplifications when H ≪ N 1 2 ), but the restriction stems from a result on square-free smo oth in tegers in short in terv als; see Lemma 3.4. Since we are only proving a lo wer b ound and therefore can pick con venien t scales to w ork with, it is lik ely p ossible to circumv en t this using the Matom¨ aki–Radziwi l l machinery; see the w ork of Jain [31] and the references within. Ho w ever, w e ha v e opted for simplicit y , esp ecially since our result captures the most interesting regime where there is a transition of the large fluctuations from √ log N down to √ log log N . 2 It w ould b e interesting to prov e the corresp onding almost sure upp er b ound in any regime of H . F or rather large H , it is lik ely that the sophisticated ideas in [34, 36, 18, 1] relating the short sum P N − H 1 and b ≥ 0 , wher e ∥ Z ∥ p = ( E | Z | p ) 1 p . Then for q such that 1 /p + 1 /q = 1 , we have sup ∥ u ∥ Lip ≤ 1 | E uX − E uY | ≪ b 1 − 2 / ( q k +2) sup | θ | =1 Z ∞ −∞      P k X l =1 θ l X l ≤ x ! − P k X l =1 θ l Y l ≤ x !      d x ! 2 / ( kq +2) . Her e the supr emum is taken over al l functions u : R k → R with Lipschitz semi-norm at most 1 . This is a more direct approac h compared to the use of Stein’s metho d via exchangeable pairs in the w ork of Harp er [21, 24]. In particular, the author was unable to adapt the latter metho d to handle the complex dep endencies betw een large primes p and q suc h that p | n and q | n + 1 when dealing with P n ≤ N f ( n ( n + 1)), say . The adv antage of our approac h is that we obtain a quantitativ e multiv ariate cen tral limit theorem purely from the one-dimensional martingale structure. See also the recent work of Kow alski and Un trau [33], Humphries [30], as well as earlier w ork of Saksman and W ebb [41] on the use of the Kantoro vich–W asserstein distance in n um b er theory . Equipp ed with a quantitativ e m ultiv ariate central limit theorem, w e may compare the dis- tribution of our k sums to the distribution of k Gaussian random v ariables with a prescrib ed co v ariance structure go v erned by the arithmetic of the set A . Given this information, w e may readily establish the almost sure low er b ound by an application of Prop osition 2.2 and the Borel-Can telli lemma. 2.4. Arithmetic input. After applying Prop osition 2.1 to P l θ l |A N l | − 1 2 P n ∈A l f ( n ) for Rade- mac her (or Steinhaus) f , what w e essen tially hav e to con trol is the num b er of solutions to the equation n 1 n 2 n 3 n 4 = □ (or n 1 n 2 = n 3 n 4 ) with n i ∈ A N l i for 1 ≤ i ≤ 4 coming from (at most) four differen t scales. Henceforth w e will refer to this as the fourth moment e quation . In each of the examples w e consider in the pap er, w e would like for this equation to “only” (in an asymptotic sense) ha v e the trivial solutions where the n i are equal in pairs, con tributing appro ximately N l i N l j solutions for 1 ≤ i, j ≤ 4. This means that w e would like to rule out the man y non-trivial solutions that o ccur in the full ranges of integers. In the following subsections, w e will describ e how we achiev e this, as well as the subtleties in the previously describ ed method for the examples w e consider in this pap er. 2.4.1. Polynomial images. This case will b e our most streamlined application of the metho d as describ ed ab ov e. W e will utilize the main results of [32, 9], giving a p o wer-sa ving b ound on the n um b er of non-trivial solutions to the fourth momen t equations P ( n 1 ) P ( n 2 ) P ( n 3 ) P ( n 4 ) = □ (or P ( n 1 ) P ( n 2 ) = P ( n 3 ) P ( n 4 )) with all n i ≤ N . This allows us to choose the N l for 1 ≤ l ≤ k so that we can con trol the co v ariances of the sums N − 1 2 l P n ≤ N l f ( P ( n )) quite crudely , b y simply b ounding the n um b er of non-trivial solutions to the equation with n i ranging up to the maxim um of the N l i . Ho w ever, the martingale structure that w e use forces us to com bine the p o wer-sa ving b ound ab o ve with upper b ounds for (very) smo oth v alues of p olynomial images, resulting in a quan- titativ e bound of the form exp  − √ log X  in the martingale cen tral limit theorem (where X is so that log N l ≍ log X for all 1 ≤ l ≤ k ); see Corollary 5.2. Nonetheless, this is certainly strong LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 7 enough to allo w us to access the tails that o ccur with probabilit y ≈ (log X ) − C 2 / 2 , leading to the almost sure upp er b ound as describ ed b efore. R emark 2.3 . One could also obtain the almost sure upp er b ound in the Steinhaus case b y ev aluating the 2 k -th momen t of P n ≤ N f ( P ( n )) for k ≈ log log N , as w as done in the work of W ang and Xu [43]. A careful insp ection of their pro of shows that this is indeed the uniformity they obtain. This appears harder to do in the Rademacher case. 3 F or the low er b ound, passing to the multiv ariate central limit theorem will result in a further loss, namely a bound of the form √ k exp  − √ log X k  for the approximation; see Corollary 5.5. Ho w ever, it will be sufficien t to choose k = (log X ) ε 0 for some small ε 0 > 0 in order to create large fluctuations of order ≈ √ 2 log k ≈ √ log log X almost surely . 2.4.2. Inte gers in a short interval. A ttempting to follo w the strategy as in the p olynomial case here runs into the follo wing difficulties. Firstly , the b ounds we hav e for the num b er of non- trivial solutions to n 1 n 2 n 3 n 4 = □ (or n 1 n 2 = n 3 n 4 in the Steinhaus case) are muc h w eak er and get w orse as H gets bigger. Secondly , ev en when H = N α in whic h case w e hav e p ow er-sa ving b ounds on the num b er of non-trivial solutions, as b efore, the very smo oth solutions (including the trivial ones) yield a martingale central limit theorem with a quantitativ e b ound of shap e roughly exp  − √ log X  . This is not enough to capture the large fluctuations of size C √ log X that occur probability roughly X − C 2 / 2 . R emark 2.4 . When k = 1, the abov e strategy nonetheless yields a significan t quan titativ e impro v ement of the cen tral limit theorem of Chatterjee and Soundarara jan [8], but w e do not state this in the pap er. W e note that the result of Soundarara jan and Xu [42] is also quantitativ e, but stated in a different form in terms of the characteristic function, with an apparen t large loss of e t 2 / 2 . Ho wev er, it app ears that by b eing less crude with the b ound | 1 + it | ≤ e t 2 / 2 for large t in their w ork, applying Esseen’s inequalit y (truncated F ourier inv ersion) to their result w ould probably recov er a quantitativ e central limit theorem of similar strength. T o bypass these issues, w e condition on the v alues of f on the small primes, and approximate the sums P N l − H l 2 log 2 − 1; see Proposition 6.4. F or us, ho w ev er, it is crucial that w e obtain a sufficiently strong quan titative b ound for the num b er of such integers; see Lemmas 3.3 and 6.1. 3 W e thank Christopher Atherfold for discussions p ertaining to this remark. 8 BESFOR T SHALA 2.4.3. Other examples and limitations. The metho d used in this pap er may be used to pro v e almost sure lo wer b ounds on large fluctuations of other examples, such as: shifted primes, sums of squares in short interv als [42], rough integers [45], and exp onential sums with random m ultiplicativ e co efficients P n ≤ N f ( n ) e ( nθ ) [3, 19, 42]. Ho w ev er, we are not able to handle the case of restricted num b er of prime factors ω ( n ) ≤ w = o (log log N ) for n ≤ N considered in [29, 22], for the num b er of non-trivial solutions to the fourth momen t equation there is very large. Even when w is finite, the sa ving o v er the n um b er of trivial solutions is at most double- logarithmic in N , resulting in a very small c hoice for the n um b er of scales k that w e can hope to use. It would b e interesting to determine whether this is a reflection of the truth, that is, whether the large fluctuations of P n ≤ N , ω ( n ) ≤ w f ( n ) differ from what would b e expected from the appro ximate Gaussian distribution. In another direction, it would also be interesting to prov e almost sure upp er b ounds for other examples, suc h as shifted primes. In con trast with short interv als, say , it is not clear ho w one w ould pro ceed here, as this sum lik ely cannot b e related to a random Euler pro duct — this is an ongoing inv estigation. 3. A uxiliar y Resul ts In this section, we collect and/or prov e some auxiliary results that w e will use throughout our proofs. 3.1. Num b er theory results. W e start b y stating a result of Nair and T enenbaum [37] (see also the w ork of Henriot [26]), in a less general form that will b e sufficient for us. This is a generalization of a well-kno wn lemma of Shiu. Prop osition 3.1. L et F b e a non-ne gative multiplic ative function such that ther e exists a c onstant A ≥ 1 and ϵ > 0 such that F ( m ) ≤ Am ϵ for al l m ∈ N . L et Q ∈ Z [ x ] b e a fixe d p olynomial of de gr e e d with no fixe d prime factor, and let ρ ( n ) b e the numb er of solutions to Q ( x ) ≡ 0 (mo d n ) . Supp ose that Q has r distinct irr e ducible factors Q 1 , Q 2 , . . . , Q r , and write Q = Q r i =1 Q γ i i . L et ρ i ( n ) b e the numb er of solutions to Q i ( x ) ≡ 0 (mo d n ) for 1 ≤ i ≤ r . Denote the discriminant of the squar e-fr e e kernel of Q by D . If ϵ < 1 8 d 2 , then X N 0 such that X N 0. Supp ose Q has r distinct irreducible factors with the largest degree of an irreducible factor b eing d . Applying Proposition 3.1 to τ 3 and the p olynomial Q , and using the subm ultiplicativity of τ 3 , w e obtain X N 0 and let N and H b e lar ge enough such that H ≥ N 5 ε . L et Ω( n ) denote the numb er of prime factors of n , c ounte d with multiplicity. Then ther e exists ε ′ > 0 such that # { N − H < n ≤ N : Ω( n ) > (1 + ε ) log log N } ≪ H (log N ) ε ′ . Pr o of. The quantit y we wan t to b ound is certainly at most 1 exp((1 + ε ) log log N log (1 + ε )) X N − H 1 2 , and let N ≥ H = H ( N ) ≥ N 4 α +1 5 . The numb er of squar e-fr e e inte gers in the short interval ( N − H , N ] such that P + ( n ) > N α is ≫ H + O  H log N  . Pr o of. The num b er of such n is at least X N α

N 2 l , say . F or a fixed l , the probability that (3.1) fails for all N ∈ ( X l , X 2 l ] and a suitably small implied constant is at most E ( X l ) b y the condition of the prop osition. Now w e may choose a subsequence N l k of N l so sparse dep ending on E suc h that P ∞ k =1 E ( N l k ) conv erges. Then b y Borel-Cantelli, (3.1) fails for at most finitely many l k . □ Finally , w e record a quantitativ e version of Slepian’s lemma. Lemma 3.8 ([24], Normal Comparison Result 1) . Supp ose that k ≥ 2 , and that ϵ ≥ 0 is sufficiently smal l (i.e., less than a c ertain smal l absolute c onstant). L et Y 1 , Y 2 , . . . , Y k b e me an zer o, varianc e one, jointly normal r andom variables, and supp ose E Y l 1 Y l 2 ≤ ϵ whenever l 1  = l 2 . Then, for any 100 ϵ ≤ δ ≤ 1 / 100 , we have P  max 1 ≤ l ≤ k Y l ≤ p (2 − δ ) log k  ≤ exp − Θ k δ / 20 √ log k !! + k − δ 2 / (50 ϵ ) . 4. General Framew ork W e are going to deduce our results from the follo wing general framework. Throughout the rest of this paper, let f b e a Rademacher (or Steinhaus) random multiplicativ e function. W e are interested in subsets A N ⊆ { 1 , 2 , . . . , N } of square-free (or all) p ositive integers, sampled at v arious N 1 , N 2 , . . . , N k , with (some of ) the following prop erties for any 1 ≤ l 1 , · · · , l 4 ≤ k : (I) There exists ε 1 > 0 such that the n um b er of non-trivial 5 solutions to n 1 n 2 n 3 n 4 = □ (or n 1 n 4 = n 2 n 3 ) with n i ∈ A N l i and P + ( n 1 ) = P + ( n 2 ) , P + ( n 3 ) = P + ( n 4 ) is ≤ ε 1 q |A N l 1 | · · · |A N l 4 | . (I’) There exists ε ′ 1 > 0 such that the num b er of non-trivial solutions to n 1 n 2 n 3 n 4 = □ (or n 1 n 4 = n 2 n 3 ) with n i ∈ A N l i and P + ( n 1 ) = · · · = P + ( n 4 ) is ≤ ε ′ 1 q |A N l 1 | · · · |A N l 4 | . (Note that ε ′ 1 ma y b e tak en to b e at most ε 1 if (I) is satisfied.) (I I) There exists ε 2 > 0 such that the n umber of n 1 ∈ A N l 1 and n 2 ∈ A N l 2 suc h that P + ( n ) = P + m is ≤ ε 2 |A N l 1 ||A N l 2 | . (I I I) W e either ha ve: (I I I a ) If N l 1 ≤ N l 2 then A N l 1 ⊆ A N l 2 , or (I I I b ) If l 1  = l 2 , then A N l 1 ∩ A N l 2 = ∅ . R emark 4.1 . Condition (I II) is automatic if w e begin with an infinite subset of square-free (or all) positive integers A , and let A N = A ∩ ( N − H , N ] for some H = H ( N ) → ∞ , as long as the scales N 1 , N 2 , . . . , N k are sufficien tly separated. Let (4.1) S N = 1 p |A N | X n ∈A N T f ( n ) = X p 1 p |A N | X n ∈A N P + ( n )= p T f ( n ) =: X p M p,N , where here and throughout the pap er T denotes simply the iden tity in the Rademac her case, and √ 2 times either the real or imaginary part in the Steinhaus case. 5 W e call a solution non-trivial if the n i are not equal in pairs. 12 BESFOR T SHALA 4.1. Appro ximate joint Gaussianity. F or any sequence of integers N 1 < N 2 < · · · < N k and real co effic ien ts c 1 , c 2 , . . . , c k with P k l =1 | c l | 2 = 1, the sum c 1 S N 1 + c 2 S N 2 + · · · + c k S N k = X p k X l =1 c l M p,N l is a martingale, with P k l =1 c l M p,N l b eing a martingale difference sequence indexed b y primes p . Applying Proposition 2.1 (or Prop osition 3.5) to P k l =1 c l M p,N l leads us to b ounding the quan tities A := X p E      k X l =1 c l M p,N l      4 and B := E       X p      k X l =1 c l M p,N l      2 − 1       2 . In the Steinhaus case, for our particular martingale, the last term from Prop osition 3.5 is ≪ A . This is b ec ause upon expanding E       X p   k X l =1 c l M p,N l ! 2 + k X l =1 c l M p,N l ! 2         2 , for all primes p  = q w e ha v e E M 2 p,N l 1 M 2 q ,N l 2 = E M 2 p,N l 1 M 2 q ,N l 2 = E M 2 p,N l 1 M 2 q ,N l 2 = E M 2 p,N l 1 M 2 q ,N l 2 = 0 , whereas for the primes p = q the surviving terms are (note that E  P k l =1 c l M p,N l  4 = E  P k l =1 c l M p,N l  4 = 0 since E f ( p ) 4 = 0) ≪ X p E k X l =1 c l M p,N l ! 2 k X l =1 c l M p,N l ! 2 = A. 4.1.1. Bounding A . W e ha v e A = X p X 1 ≤ l i ≤ k c l 1 · · · c l 4 E M p,N l 1 · · · M p,N l 4 . F urther, E M p,N l 1 · · · M p,N l 4 = 1 q |A N l 1 | · · · |A N l 4 | X n i ∈A N l i P + ( n i )= p E f ( n 1 ) · · · f ( n 4 ) . Note that E f ( n 1 ) · · · f ( n 4 ) is non-zero only if n 1 n 2 n 3 n 4 is a p erfect square (or n 1 n 2 = n 3 n 4 ), in which case the exp ectation is equal to 1. Therefore, b y moving the sum ov er p inside, w e ha v e A = X 1 ≤ l i ≤ k c l 1 q |A N l 1 | · · · c l 4 q |A N l 4 | X n i ∈A N l i n 1 n 2 n 3 n 4 = □ (or n 1 n 2 = n 3 n 4 ) P + ( n i )= P + ( n j ) ∀ i,j 1 . LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 13 The con tribution from the trivial solutions (note that b y condition (I I I), n 1 = n 2 forces one of A N l 1 , A N l 2 to be a subset of the other, and similarly for the other v ariables) is ≪ X 1 ≤ l i ≤ k | c l 1 | · · · | c l 4 | q |A N l 1 | · · · |A N l 4 | X n 1 ∈A N l 1 ∩A N l 2 n 3 ∈A N l 3 ∩A N l 4 P + ( n 1 )= P + ( n 3 ) 1 ≪ ε 2 k 2 , b y condition (I I). Here we used that if, sa y A N l 1 ∩ A N l 2 = A N l 1 , then certainly |A N l 2 | ≥ |A N l 1 | , and then applied the Cauc h y-Sc hw arz inequalit y for the sum o v er the l i . Note that if (I I I b ) is satisfied, then n 1 = n 2 forces l 1 = l 2 (and similarly for the other v ariables), in which case the con tribution from the trivial solutions w ould b e ≪ ε 2 P 1 ≤ l 1 ,l 3 ≤ k | c l 1 | 2 | c l 3 | 2 ≪ ε 2 . The con tribution from non-trivial solutions is ≪ ε ′ 1 X 1 ≤ l i ≤ k | c l 1 | · · · | c l 4 | ≪ ε ′ 1 k 2 using P k l =1 | c l | ≪ √ k b y Cauch y-Sch warz and condition (I’). W e conclude that (4.2) A ≪ k 2 ( ε ′ 1 + ε 2 ) , or (4.3) A ≪ k 2 ε ′ 1 + ε 2 if (II I b ) is satisfied. 4.1.2. Bounding B . W e hav e B = 1 + X p,q X 1 ≤ l i ≤ k c l 1 q |A N l 1 | · · · c l 4 q |A N l 4 | X n i ∈A N l i n 1 n 2 n 3 n 4 = □ ( n 1 n 4 = n 2 n 3 ) P + ( n 1 )= P + ( n 2 )= p P + ( n 3 )= P + ( n 4 )= q 1 − 2 X p X 1 ≤ l i ≤ k c l 1 q |A N l 1 | c l 2 q |A N l 2 | X n i ∈A N l i n 1 n 2 = □ (or n 1 = n 2 ) P + ( n 1 )= P + ( n 2 )= p 1 Note that n 1 n 2 = □ if and only if n 1 = n 2 in the Rademac her case, th us we hav e that the last sum is equal to − 2 X 1 ≤ l i ≤ k c l 1 q |A N l 1 | c l 2 q |A N l 2 | min {|A N l 1 | , |A N l 2 |} = − 2 + O   X 1 ≤ l 1 P k X l =1 c l M p,N l is a martingale (in fact it is a w eigh ted sum of the indep endent random v ariables f ( p ) with p > P ), where (4.6) V c = v u u u t e E   X p>P k X l =1 c l M p,N l   2 = s X l 1 ≤ l 2 c l 1 c l 2 e E S N l 1 S N l 2 . Here and throughout e E denotes conditional exp ectation b eing taken only o v er primes > P . Theorem 4.3. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function, and supp ose the A N l for 1 ≤ l ≤ k satisfy c onditions (I’), (II), (III), and (IV), with the values ε ′ 1 , ε 2 , and P , r esp e ctively. With pr ob ability (4.7) 1 − O k q ( ε ′ 1 + ε 2 ) + (1 − 1 (III b ) ) k max 1 ≤ l 1 P k X l =1 c l M p,N l ≤ x   − 1 p 2 π V 2 c Z x −∞ e − t 2 2 V 2 c d t       ≪  k p ε ′ 1 + ε 2  1 10 1 + | x | 16 5 , wher e e P denotes c onditional pr ob ability b eing taken only over f ( p ) for primes p > P . Pr o of. W e b egin by applying Prop osition 2.1 to the martingale 1 V c X p>P k X l =1 c l M p,N l . Noting that 1 V 2 c e E X p>P k X l =1 c l M p,N l ! 2 − 1 = 0 b y the definition of V c , after applying the c hange of v ariables V c x 7→ x , we hav e       e P   X p>P k X l =1 c l M p,N l ≤ x   − 1 p 2 π V 2 c Z x −∞ e − t 2 2 V 2 c d t       ≪  V 12 c · e E P p>P    P k l =1 c l M p,N l    4  1 5 1 + | x | 16 5 . 16 BESFOR T SHALA No w, b y exactly the same computations leading to Theorem 4.2, w e ha v e E e E X p>P      k X l =1 c l M p,N l      4 ≪ k 2 ( ε ′ 1 + ε 2 ) , and E V 2 c ≪ 1 + (1 − 1 (II I b ) ) k max 1 ≤ l 1 P      k X l =1 c l M p,N l      4 ≤ k q ε ′ 1 + ε 2 and V 2 c ≪  k q ε ′ 1 + ε 2  − 1 / 12 hold, th us finishing the pro of. □ The adv antage of introducing condition (IV) and Theorem 4.3 is that the parameters ε ′ 1 and ε 2 can b e tak en to b e muc h smaller than b efore (depending on the size of P ). This leads to a v ery precise approximation of the conditional probabilities ov er primes bigger than P , at least o v er most realizations of the f ( p ) for p ≤ P . 4.2. Multiv ariate Central Limit Theorem. Finally , we use Prop osition 2.2 to b o otstrap Theorems 4.2 and 4.3 to quan titativ e m ultiv ariate cen tral limit theorems. Theorem 4.4. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function, and supp ose the A N l for 1 ≤ l ≤ k satisfy c onditions (I), (I’), (II) and (III), with the values ε 1 , ε ′ 1 , ε 2 , r esp e ctively. F urther, let X = ( S N 1 , S N 2 , . . . , S N k ) (r e c al l (4.1) for the definition of S N ) and let Y b e a standar d multivariate Gaussian r andom ve ctor with indep endent c o or dinates. Then (4.8) sup ∥ u ∥ Lip ≤ 1 | E uX − E uY | ≪ k 1 2 k 2 ε 2 + k max 1 ≤ l 1 P k X l =1 c l M p,N l is conditionally a weigh ted sum of indep enden t random v ariables. Therefore, instead of pro ving Theorem 4.3 and b o otstrapping it to Theorem 4.4, we could ha v e opted to use the Normal Appro ximation Result 1 from the w ork of Harp er [24], which yields a stronger result in terms of the dep endence on k . How ev er, our argument only utilizes the martingale structure and ma y p oten tially b e useful in other situations with a weak er assumption than (IV). Theorem 4.4 may then b e used to estimate the probabilit y that max 1 ≤ l ≤ k S N l ≤ t in terms of the probability that max 1 ≤ l ≤ k Y l ≤ t + O (1). Corollary 4.6. A dapt the notation and c onditions in the statement of The or em 4.4. Then P  max 1 ≤ l ≤ k S N l ≤ t  ≤ P  max 1 ≤ l ≤ k Y l ≤ t + η  + O   k 3 2 η − 1 k 2 ( ε 1 + ε ′ 1 + ε 2 ) + k max 1 ≤ l 1 t + η . Note that the Lipschitz constant of u is b ounded b y k η − 1 , so after rescaling u b y k η − 1 , w e get an extra k η − 1 in the error. Observe that the difference b etw een the desired probabilities is b ounded by E u ( S N 1 , . . . , S N k ) − E uY , thus finishing the pro of. □ 4.3. Slo w v ariation. Here w e shall prov e a general slow v ariation prop erty of sums of random m ultiplicativ e functions, which is also applicable in short interv als. This generalizes a result found in the work of Lau, T enen baum and W u [34], making use of Lemma 3.6. In this pap er, w e will use this prop osition in pro ving the almost sure upp er b ound in the case of p olynomial images, but it is relev ant in pro ving almost sure upp er b ounds in other cases to o. Prop osition 4.7. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function. L et A b e an infinite subset of the squar e-fr e e (or al l) p ositive inte gers, and supp ose that A N = A ∩ ( N − H , N ] for some N ≥ H = H ( N ) → ∞ as N → ∞ . F or any incr e asing se quenc e 6 One p ossibility is s ( x ) = ϕ ( t + η − x ) / ( ϕ ( t + η − x ) + ϕ ( x − t )) with ϕ ( x ) = exp( − 1 /x ) for x > 0 and ϕ ( x ) = 0 for x ≤ 0. 18 BESFOR T SHALA ( N l ) ∞ l =1 such that N 1 is lar ge enough and N l +1 − H l +1 ≤ N l for al l l , we have P   max N l 0 } . Note that |A N | ∼ κ P N for some κ P > 0 dep ending 8 on P . Recall (5.1) S N = 1 p |A N | X n ∈A N T f ( n ) ∼ 1 √ κ P N X n ≤ N T f ( P ( n )) . Moreo v er, for a large parameter X , let N l = λ l X with λ = exp  √ log X  , where 1 ≤ l ≤ k = (log X ) ε 0 with ε 0 > 0 to b e sp ecified later. W e b egin b y chec king that these sets satisfy prop erties (I), (I’), (I I) and (I I I). (I). W e employ the result of [9] ([32]), whic h w e recall here. Prop osition 5.1. L et P b e a p olynomial as in the statement of The or em A. Ther e exists a c onstant δ P > 0 such that the numb er of non-trivial solutions to the e quation P ( n 1 ) P ( n 2 ) P ( n 3 ) P ( n 4 ) = □ ( or P ( n 1 ) P ( n 2 ) = P ( n 3 ) P ( n 4 )) with n i ≤ N is ≪ N 2 − δ P . Using the ab ov e with N = max { N l 1 , . . . , N l 4 } ≪ λ k X allo ws us to take ε 1 ≪ X − δ ′ P for some sligh tly smaller fixed δ ′ P > 0 (p ossibly dep ending on P ), since λ is so that λ k ≪ ϵ X ϵ for all ϵ > 0. (I’). W e will simply tak e ε ′ 1 = ε 1 . (II). W e chec k condition (I I) as follo ws. W e in tro duce a smo othness parameter y , and split the range of P + ( n 1 ) = P + ( n 2 ) = p to p ≤ y and p > y . The latter range also contains very large primes, which are dealt with separately . In any case, we sho w the con tribution of primes p > y is bounded ab ov e by X p>y |A N l 1 ||A N l 2 | p 2 ≪ |A N l 1 ||A N l 2 | y . The contribution of primes p ≤ y is b ounded by the pro duct of the num b er of y -smo oth v alues of n ∈ A N and the n umber of y -smooth v alues of m ∈ A M . Optimizing the choice of y yields the sa ving ε 2 . Concretely , using the calculations from the end of [9, Section 4] ([32, Section 2]), we may tak e ε 2 ≪ 1 y + ψ P ( N l 1 , y ) ψ P ( N l 2 , y ) N l 1 N l 2 , where ψ P ( N , y ) is the n um b er of n ≤ N such that P ( n ) is y -smo oth. By a result of Hmyro v a [27], when y ≥ log N w e hav e the b ound 9 ψ P ( N , y ) ≪ N ( e/u ) u , where u = log N / log y . After an unpleasant exercise in manipulating logarithms and optimizing in y , w e choose y = exp  √ log X log log X  . This giv es ε 2 ≪ exp( − √ log X log log X ). (III). Note that the sets A N are nested, so (I I I a ) is satisfied. 8 In the Steinhaus case w e hav e κ P = 1, whereas in the Rademac her case, κ P is the density of square-free v alues of P . 9 The b ound there is stated for irreducible P , but obviously the n umber of v alues of n ≤ N for which P ( n ) is y -smo oth is b ounded ab ov e by the num b er of n ≤ N for which some irreducible factor of P ( n ) is y -smo oth. 20 BESFOR T SHALA 5.1. Almost sure upp er b ound. W e no w ha ve everything w e need to pro v e the almost sure upp er b ound. W e first apply Theorem 4.2 with k = 1 (at a single scale), yielding the following quan titativ e cen tral limit theorem. Corollary 5.2. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function, and let P ∈ Z [ x ] b e a p olynomial as in The or em A. We have       P   1 √ κ P N X n ≤ N T f ( P ( n )) ≤ x   − 1 √ 2 π Z x −∞ e − t 2 2 d t       ≪ 1 1 + | x | 16 5 exp  − 1 5 p log N log log N  . W e will use Corollary 5.2 at certain test p oints ( N l ) ∞ l =1 , com bined with the following almost sure slo w v ariation result b etw een the test p oints. Lemma 5.3. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function, and let P ∈ Z [ x ] b e a p olynomial as in The or em A. F or any c onstant A > 0 , ther e exists a c onstant c > 0 such that for N l = ⌊ e l c ⌋ , we have that max N l 0 dep ending on P . T aking x = p N l +1 / (log N l +1 ) A , w e obtain P   max N l 0 suc h that for all l ∈ N we hav e (5.2)       X n ≤ N l f ( P ( n ))       ≤ C p N l log log N l , where N l is as in the statement of the lemma (in the Steinhaus case, w e do this separately for the real and imaginary parts). By Corollary 5.2, the probabilit y that the ab o ve fails for a fixed LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 21 l and large enough implied constant C is ≪ (log N l ) − C 2 2 . Making C larger (in terms of c ) if necessary , w e ha ve that the series of probabilities ∞ X l =1 (log N l ) − C 2 2 = ∞ X l =1 1 l cC 2 2 < ∞ , th us (5.2) holds almost surely by Borel-Cantelli (with a p ossibly even larger constant C to accoun t for the finitely many failures). 5.2. Almost sure lo w er bound. Here w e will use Corollary 4.6 with the scales N 1 , N 2 , . . . , N k defined at the start of the section. In fact, w e will prov e something slightly stronger. Theorem 5.4. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function and let P ∈ Z [ x ] b e a p olynomial as in The or em A. Ther e exists a c onstant c > 0 such that for any lar ge enough X , we have that max X 0 is sufficiently small. □ No w the probabilit y P  max 1 ≤ l ≤ k Y l ≪ √ log k  can b e suitably estimated using Lemma 3.8. Recall that E Y l 1 Y l 2 = 0 for i  = j since the comp onents of Y are indep endent, so we ha ve P  max 1 ≤ l ≤ k Y l ≤ p (2 − δ ) log k  ≤ exp − Θ k δ / 20 √ log k !! . Recalling that k = (log X ) ε 0 and taking η > 0 and δ > 0 fixed, we obtain P  max 1 ≤ i ≤ k S N l ≫ p log log X  ≥ 1 − O  exp  − (log X ) O ( ε 0 )  for a suitably small implied constant (making ε 0 > 0 smaller if necessary). Theorem 5.4 follo ws b y noticing that X ≤ λX ≤ N l ≤ λ k X ≤ exp  (log X ) 1 2 + ε 0  X ≤ X 2 for all 1 ≤ l ≤ k . 22 BESFOR T SHALA 6. Proof of Theorem B: Shor t inter v als Here we will require the more sophisticated part of Corollary 4.6. Let H b e as in the statement of Theorem B and consider A N = { n ∈ N : N − H < n ≤ N } . Note that |A N | ∼ κH for some 10 κ > 0. Recall S N = 1 p |A N | X n ∈A N T f ( n ) ∼ 1 √ κH X N − H 0 dep ending on the function H and then even smaller ε 0 > 0, find k = ( X/H ( X )) ε 0 primes l in the range h ( X ) / 2 < l ≤ h ( X ), where h ( X ) := ( X/H ( X )) δ . W e will not fix δ or ε 0 for no w, as there will b e v arious (but finitely man y) places where w e ma y ha ve mak e them smaller. F or these v alues of l , let N l = l X with corresp onding H l = H ( N l ). Observe that for large enough X and small enough δ > 0, the sets A N l are disjoin t. It is not these sets that we will apply Theorem 4.3 to. If we did this, the v ery smo oth in tegers in the sets A N l w ould prev ent us from obtaining a suitably small v alue for ε 2 . W e b ypass this b y splitting S N l as (6.1) X N l − H l N 1 / (log N 1 ) 2 (hence since H is increasing we hav e H l ≫ X h ( X ) / (log X ) 2 for all v alues of l ), we c ho ose some ε > 0 to b e sp ecified later and additionally split the first part as (6.2) X N l − H l (1+ ε ) log log X f ( n ) . W e first show that the sum ov er in tegers with Ω( n ) > (1 + ε ) log log X ab o ve ma y b e typically ignored for all v alues of l . Lemma 6.1. Assume that H 1 > N 1 / (log N 1 ) 2 and let ε > 0 . Ther e exists ε ′ = ε ′ ( ε ) such that with pr ob ability 1 − O ((log X ) − ε ′ +2 ε 0 ) we have 1 √ H l X N l − H l (1+ ε ) log log X f ( n ) ≪ 1 for al l of the k chosen values of h ( X ) / 2 < l ≤ h ( X ) . Pr o of. The probability that         X N l − H l (1+ ε ) log log X f ( n )         ≥ λ is b ounded by λ − 2 times the num b er of n ∈ ( N l − H l , N l ] with Ω( n ) > (1 + ε ) log log X by Mark o v’s inequality using a second moment estimate. The num b er of such n is ≪ H l / (log X ) ε ′ b y Lemma 3.3. T aking a union b ound o v er k v alues of l and recalling that k = ( X/H ( X )) ε 0 ≪ (log X ) 2 ε 0 (making ε 0 > 0 smaller if necessary) giv es a probabilit y 1 − O ((log X ) − ε ′ +2 ε 0 ) that all of the sums are b ounded. □ 10 In the Steinhaus case we hav e κ = 1, whereas in the Rademacher case κ = 6 π 2 . LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 23 If H 1 ≤ N 1 / (log N 1 ) 2 (in whic h case by concavit y of the function H w e ha v e H l ≪ X h ( X ) / (log X ) 2 for all v alues of l ), let B N l b e the subset of A N l of those integers n with P + ( n ) > ( X h ( X )) 2 3 . If H 1 > N 1 / (log N 1 ) 2 , add the restriction Ω( n ) ≤ (1 + ε ) log log X . Now w e w ork with X n ∈B N l f ( n ) + O               X N l − H l (1+ ε ) log log X f ( n ) + O  p H l (log log X ) 1 3  for all go o d l . (The second sum app ears only when H 1 > N 1 / (log N 1 ) 2 .) 6.1. Almost sure lo w er b ound. Throughout this section, all v alues of l will be go o d even if w e do not state so. As b efore, we will actually prov e some thing sligh tly stronger. Theorem 6.3. L et f b e a R ademacher (or Steinhaus) r andom multiplic ative function and let H b e a function as in The or em B. F or any lar ge enough X , we have that max X ( X h ( X )) 2 3 , then n ′ i = n i /p satisfy n ′ 1 n ′ 2 n ′ 3 n ′ 4 = □ (or n ′ 1 n ′ 2 = n ′ 3 n ′ 4 ) and n ′ i ≤ N l i /p ≤ X h ( X ) /p . F or a fixed p , the num b er of such n ′ i is ≪ ( X h ( X ) log X/p ) 2 . Ov er all primes p > ( X h ( X )) 2 3 , w e hav e at most ≪ ( X h ( X ) log X ) 2 X ( X h ( X )) 2 3

0 such that E e Y l 1 e Y l 2 ≤ ϵ whenever l 1  = l 2 . The heart of the matter lies in the following prop osition. Prop osition 6.4. Assume that H is as in The or em B. L et l 1 and l 2 b e primes in the r ange h ( X ) / 2 < l 1 , l 2 ≤ h ( X ) = ( X/H ( X )) δ . The numb er of non-trivial n 1 , n 3 ∈ B N l 1 and n 2 , n 4 ∈ B N l 2 with P + ( n 1 ) = P + ( n 2 ) and P + ( n 3 ) = P + ( n 4 ) such that n 1 n 2 n 3 n 4 = □ (or n 1 n 4 = n 2 n 3 ) is ≪ H l 1 H l 2 /h ( X ) 2 if δ > 0 and ε > 0 ar e smal l enough. 11 11 Recall the definition of B N l for the dep endence on ε . LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 25 Pr o of. W rite γ l = H l / N l so that H l = γ l N l . Notice that γ l 1 ≍ γ l 2 ≍ γ = γ ( X ) = H ( X h ( X )) / ( X h ( X )) uniformly in the v alues of l 1 , l 2 , since H is increasing and concav e. Without loss of generality , w e ma y assume that l 1 ≤ l 2 . W e b egin with the Steinhaus case. W e utilize the parametization of the solutions to n 1 n 4 = n 2 n 3 as n 1 = g a, n 2 = g b, n 3 = ha, n 4 = hb, where ( a, b ) = 1 and g = ( n 1 , n 2 ) , h = ( n 3 , n 4 ) . In particular, P + ( n 1 ) | g and P + ( n 3 ) | h , so b oth g , h > ( X h ( X )) 2 3 . Since we are concerned with non-trivial solutions, w e ma y assume that g  = h and a  = b . Note that (1 − γ l 1 ) 2 = l 2 2 N 2 l 1 (1 − γ l 1 ) 2 l 2 1 N 2 l 2 ≤ l 2 2 n 1 n 3 l 2 1 n 2 n 4 = ( l 2 a ) 2 ( l 1 b ) 2 ≤ l 2 2 N 2 l 1 l 2 1 N 2 l 2 (1 − γ l 2 ) 2 = 1 (1 − γ l 2 ) 2 , th us (6.4) γ ≪     l 2 a l 1 b − 1     ≪ γ . In particular, if l 2 a  = l 1 b , then we hav e b oth l 2 a ≫ γ − 1 and l 1 b ≫ γ − 1 . Since l 1 , l 2 are distinct primes and ( a, b ) = 1, the only wa y w e could hav e l 2 a = l 1 b is if a = l 1 , b = l 2 . In this case, the n um b er of c hoices for g and h is b ounded b y H l 1 H l 2 / ( l 1 l 2 ) ≪ H l 1 H l 2 /h ( X ) 2 . Throughout now w e ma y assume that l 2 a ≫ γ − 1 and l 1 b ≫ γ − 1 . Let us first handle the case when H 1 ≤ N 1 / (log N 1 ) 2 . Given a, b ≪ N 1 3 l 1 , the num b er of c hoices for g and h satisfying the constraints for the n i is certainly b ounded b y H l 1 H l 2 / ( ab ). Note that b ≫ l 2 a/l 1 b y (6.4), therefore the num b er of solutions with l 2 a, l 1 b ≫ γ − 1 is b ounded b y H l 1 H l 2 X a,b ≪ N 1 3 l 1 l 2 a,l 1 b ≫ γ − 1 (6.4) 1 ab ≪ H l 1 H l 2 X a ≪ N 1 3 l 1 a ≫ ( l 2 γ ) − 1 1 a 2 X b :(6.4) 1 ≪ H l 1 H l 2 X a ≪ N 1 3 l 1 a ≫ ( l 2 γ ) − 1 1 a 2 ( γ l 2 a/l 1 + 1) ≪ γ H l 1 H l 2 (log X + l 2 ) ≪ γ H l 1 H l 2 h ( X ) , since h ( X ) ≫ log X in this case. Accoun ting for the solutions with l 2 a = l 1 b , w e conclude that the n um b er of non-trivial solutions is b ounded b y H l 1 H l 2 /h ( X ) 2 . Let us now turn to the case when H 1 > N 1 / (log N 1 ) 2 . Let K = (1 + ε ) log log X , so that Ω( g ahb ) ≤ 2 K . The n umber of non-trivial solutions to the equation n 1 n 4 = n 2 n 3 with a  = l 1 , b  = l 2 is then (6.5) ≪ 2 2 K X a  = b γ ≪    l 2 a l 1 b − 1    ≪ γ l 2 a,l 1 b ≫ γ − 1 a,b ≪ N 1 3 l 1 2 − Ω( ab ) X g  = h g ,h ≫ N l 1 2 3 max { N l 1 (1 − γ ) /a,N l 2 (1 − γ ) /b }≤ g ,h ≤ min { N l 1 /a,N l 2 /b } 2 − Ω( g h ) . 26 BESFOR T SHALA Applying Proposition 3.1 to the sums ov er g and h yields that the righ t-hand side of (6.5) is (6.6) ≪ 2 2 K H l 1 H l 2 log X X a  = b γ ≪    l 2 a l 1 b − 1    ≪ γ l 2 a,l 1 b ≫ γ − 1 a,b ≪ N 1 3 l 1 2 − Ω( ab ) ab ≪ 2 2 K H l 1 H l 2 log X X l 2 a ≫ γ − 1 a ≪ N 1 3 l 1 2 − Ω( a ) a 2 X a  = b γ ≪    l 2 a l 1 b − 1    ≪ γ l 1 b ≫ γ − 1 2 − Ω( b ) . If l 2 a ≪ γ − 2 , we simply bound the sum o ver b b y γ l 2 a ≫ 1, namely the n umber of terms, and then the sum ov er such a is ≪ γ l 2 X l 2 a ≪ γ − 2 1 a ≪ γ l 2 log γ − 1 . If l 2 a ≫ γ − 2 , w e may apply Prop osition 3.1 to the sum ov er b , yielding that the right-hand side of (6.6) is (6.7) ≪ 2 2 K γ l 2 H l 1 H l 2 (log X ) 2 1 2        log γ − 1 + X l 2 a ≫ γ − 2 a ≪ N 1 3 l 1 2 − Ω( a ) a (log( l 2 a )) 1 2        . By a standard dyadic decomp osition for a and using Prop osition 3.1 again, w e obtain X a ≪ N 1 3 l 1 2 − Ω( a ) a (log a ) 1 2 ≪ X A =2 k ≪ X 2 1 A (log A ) 1 2 X A 0 smaller if necessary , since H ( X ) ≤ X/ (log X ) c with c > 2 log 2 − 1. Next, we consider the Rademacher case, which is very similar but a bit more tec hnical. W e utilize the parametrization of the solutions to n 1 n 2 n 3 n 4 = □ as n 1 = Ar u, n 2 = Asv , n 3 = B rv , n 4 = B su, where the v ariables A, B , r, s, u, v satisfy A = ( n 1 , n 2 ) , B = ( n 3 , n 4 ), and r , u, s, v parametrize the solutions to ( n 1 / A )( n 4 /B ) = ( n 2 / A )( n 3 /B ), as in the Steinhaus case. In particular, the v ariables r , u, s, v are all pairwise coprime. W e ma y assume that P + ( n 1 )  = P + ( n 3 ) (the contribution of suc h n i is v ery small, see the b ound ab o ve in (I’)), so that A  = B and A, B ≫ N 2 3 l 1 . Note that (1 − γ l 1 ) 2 ≤ l 2 2 n 1 n 3 l 2 1 n 2 n 4 = ( l 2 r ) 2 ( l 1 s ) 2 ≤ 1 (1 − γ l 2 ) 2 , th us (6.8) γ ≪     l 2 r l 1 s − 1     ≪ γ . In particular if l 2 r  = l 1 s then l 2 r , l 1 s ≫ γ − 1 . F urthermore, since ( r , s ) = 1, the only wa y we could hav e l 2 r = l 1 s is if r = l 1 , s = l 2 . In this case, note that m 1 = n 1 /r , m 3 = n 3 /r , m 2 = n 2 /s, m 4 = n 2 /s satisfy m 1 m 3 = m 2 m 4 and all of the m i lie in an interv al of length ≪ γ X LARGE FLUCTUA TIONS OF RANDOM MUL TIPLICA TIVE FUNCTIONS 27 cen tered at X (since N l 1 /r = X = N l 2 /s ). How ever, the num b er of choices for such m i is ≪ ( γ X ) 2 ≪ H l 1 H l 2 / ( r s ) ≪ H l 1 H l 2 /h ( X ) 2 b y our w ork abov e in the Steinhaus case, or the w ork of Soundarara jan and Xu [42]. Henceforth w e ma y assume that l 2 r  = l 1 s , th us l 2 r , l 1 s ≫ γ − 1 . Moreo v er (1 − γ l 1 )(1 − γ l 2 ) ≤ n 1 n 4 n 2 n 3 = u 2 v 2 ≤ 1 (1 − γ l 1 )(1 − γ l 2 ) , th us (6.9) γ ≪    u v − 1    ≪ γ . Again, if u  = v , we hav e u, v ≫ γ − 1 . The cases where r = s or u = v (which imply r = s = 1 or u = v = 1, resp ectiv ely , due to coprimality) simply reduce to our earlier work in the Steinhaus case. Therefore, we ma y now assume that u  = v , r  = s, l 2 r  = l 1 s and u, v , l 2 r , l 1 s ≫ γ − 1 . The n um b er of suc h solutions is b ounded by (6.10) 2 2 K X r,u,s,v l 2 r,l 1 s,u,v ≫ γ − 1 ru,r s,su,sv ≤ N 1 3 l 2 (6.8) , (6.9) 2 − Ω( rusv ) X A,B A,B ≫ N 1 3 l 2 ( N l 1 − H l 1 ) / ( ru ) ≤ A ≤ N l 1 / ( ru ) ( N l 2 − H l 2 ) / ( su ) ≤ A ≤ N l 2 / ( su ) 2 − Ω( AB ) . Prop osition 3.1 applied to the sums o v er A and B yields that the righ t-hand side of (6.10) is (6.11) ≪ 2 2 K H l 1 H l 2 log X X r,u,s,v l 2 r,l 1 s,u,v ≫ γ − 1 ru,r s,su,sv ≤ N 1 3 l 2 (6.8) , (6.9) 2 − Ω( rusv ) r su 2 . Using (6.8) w e b ound s ≫ l 2 r /l 1 ≍ r , hence the righ t-hand side of (6.11) is (6.12) ≪ 2 2 K H l 1 H l 2 log X X r,u,s,v l 2 r,l 1 s,u,v ≫ γ − 1 ru,r s,su,sv ≤ N 1 3 l 2 (6.8) , (6.9) 2 − Ω( rusv ) r 2 u 2 . Dropping all relations b et ween the pairs ( r, s ) and ( u, v ), we observ e that the b ound in (6.12) is iden tical to the square of the sum o ver a and b in (6.6), th us ma y b e bounded b y ( γ l 2 log log X ) 2 . Accoun ting for the solutions with l 2 r = l 1 s or r = s = 1 or u = v = 1, in a similar fashion as b efore, w e conclude that the num b er of non-trivial solutions is (6.13) ≪ H l 1 H l 2 h ( X ) 2 b y making δ > 0 and ε > 0 smaller if necessary . □ Applying Mark o v’s inequalit y with the second momen t and using Prop osition 6.4 gives P  E e Y l 1 e Y l 2 ≥ ϵ  ≪ h ( X ) − 2 ϵ 2 and P    E Y 2 l − 1   ≥ ϵ  ≪ h ( X ) − 2 ϵ 2 for any l 1  = l 2 and l . F or fixed and small enough ϵ > 0, w e conclude that for fixed l 1  = l 2 , w e hav e E e Y l 1 e Y l 2 ≤ ϵ and E Y 2 l ≥ 1 − ϵ with probability 1 − O ( h ( X ) − 2 ) . T aking a union b ound o v er ≍ k 2 28 BESFOR T SHALA v alues of l 1  = l 2 and l giv es that E e Y l 1 e Y l 2 ≤ ϵ and E Y 2 l ≫ 1 with probability 1 − O ( k 2 h ( X ) − 2 ). No w Lemma 3.8 gives P  max l Y l ≪ p log k  ≤ exp − Θ k O (1) √ log k !! + k − O (1) for a small enough implied constan t. Combining this with (6.3) with fixed η > 0 giv es that max l e S N l ≫ p log k holds with probability 1 − O k 3 2 η − 1  k q ε ′ 1 + ε 2  1 5( k +1) + exp − Θ k O (1) √ log k !! + k − O (1) ! . Recalling Lemmas 6.1 and 6.2, (6.1), (6.2), the v alues of ε ′ 1 , ε 2 , and that k = ( X/H ( X )) ε 0 for a suitably small ε 0 > 0, we obtain max l       1 √ H l X N l − H l