Anticoncentration of Random Sums in $\mathbb{Z}_p$

ANTICONCENTRA TION OF RANDOM SUMS IN Z p SIMONE COST A Abstract. In this pap er we in v estigate the proba bility distribution of the sum Y of ℓ indep enden t iden tically distributed random v ariables taking v alues in Z p . Our main fo cus is the regime of small v alues of ℓ , which is less explored compared to the asymptotic case ℓ → ∞ . Starting with the case ℓ = 3, we prov e that if the distributions of the Y i are uniformly b ounded b y λ < 1 and p > 2 /λ , then there exists a constan t C 3 ,λ < 1 such that max x ∈ Z p P [ Y = x ] ≤ C 3 ,λ λ. Moreo ver, when the distributions are uniformly separated from 1, the constant C 3 ,λ can be made explicit. By iterating this argument, we obtain effective anticoncen tration b ounds for larger v alues of ℓ , yielding nontrivial estimates already in small and mo derate regimes where asymptotic results do not apply . 1. Introduction The Littlewoo d–Offord problem is a classical question in probabilistic com binatorics concerning the anticoncen tration of sums of indep enden t random v ariables. It was introduced by Littlew o od and Offord in the 1940s (see [18]) in order to study the distribution of sums of random v ariables with restricted supports. In its original form ulation, giv en a list of (not necessarily distinct) in tegers, the problem asks for upp er bounds on the probability of obtaining a prescrib ed v alue by summing elemen ts from the list. Equiv alently , giv en in tegers ( v 1 , v 2 , . . . , v ℓ ), one studies the distribution of P ℓ i =1 Y i , where each Y i is uniformly distributed on { 0 , v i } (or, in an equiv alen t form ulation, on {− v i , v i } ). In [18], Littlew oo d and Offord prov ed an upp er bound of order O ( log n √ n ), which w as later impro v ed by Erd˝ os to 1 √ n (1 + o (1)) in [7]. The problem and its v arian ts arise in the analysis of random walks, random matrices, and other com binatorial structures (see for instance [4, 24]). Significant progress has been made under v arious structural assumptions on the distributions of the v ariables (see [13, 14]), as well as in settings in v olving finite groups. In particular, V aughan and W o oley [25] considered the case where the v ariables Y i are uniform on { 0 , v i } in cyclic groups, and further bounds w ere obtained by Griggs [9], Bibak [4], and Jusk evicius and Semetulskis [12]. In the latter w ork, the authors also inv estigated the case where the distributions of the Y i are uniformly b ounded b y 1 / 2. The presen t work is inspired b y these developmen ts. W e consider indep enden t random v ariables Y 1 , . . . , Y ℓ taking v alues in a cyclic group Z k , whose distributions are p oin t wise b ounded b y λ . If the distributions are uniform on a subset A ⊆ Z k with | A | = n , the problem is closely related to estimating the probability that a subset X ⊆ A of fixed cardinality ℓ has sum equal to x . This can be view ed as a v ariation of the classical Littlew oo d–Offord problem in whic h the size of the subset is fixed and the elemen ts v 1 , . . . , v n are distinct. The case of distinct integers w as studied b y Erd˝ os and later by Hal´ asz in [10] (see also [23]), w here bounds of order O ( 1 n √ n ) were obtained; v ery recent results hav e b een obtained for this problem also in Z p b y Pham and Sauermann (see [21]). 2010 Mathematics Subje ct Classific ation. 11B75, 60G50, 05D40. Key words and phr ases. Discrete F ourier T ransform, Anticoncen tration Inequalities, Probabilistic Metho ds. 1 2 These questions are also motiv ated b y applications to the set sequenceabilit y problem (see [5, 11] and [21]), where one needs an ticoncen tration estimates in finite cyclic groups in regimes where both p and ℓ are sufficiently large. In this con text, it w as brought to our attention that Lev established v ery strong asymptotic results in [15, 16]. In particular, in [16] he pro v ed that in Z p one has P [ Y = x ] ≤ 1 p + 1 n r 8 π ℓ  1 + 2 ℓ − 1 / 2 + (3 / 4) ℓ/ 2+3 ℓ 3 / 2  . Th us, for p and ℓ sufficien tly large, one recov ers the same qualitative b eha viour as in the integer setting. Ho w ev er, a direct insp ection shows that for ℓ < 24 the ab ov e estimate do es not impro v e up on the trivial bound. The aim of this paper is to address precisely this non-asymptotic regime. W e will moreov er treat distributions that are not necessarily uniform. The pap er is organized as follows. In Section 2 w e revisit the integer case. W e present a direct and self-contained proof of a b ound that app ears in [13] in a more general framework, and we mak e the constan ts explicit, impro ving in particular the case ℓ = 3. Denoting by n the cardinality of the supp ort, w e show that there exists an absolute constant D such that max x ∈ Z P [ Y = x ] ≤ D n √ ℓ − 1 . In particular, for every ϵ > 0, if ℓ is sufficien tly large with resp ect to ϵ , then max x ∈ Z P [ Y = x ] ≤ ϵ n . Moreo v er, in the case ℓ = 3, we obtain the explicit b ound ϵ = 3+1 /n 2 4 , which is already non-trivial. W e then turn to cyclic groups. When the prime factors of k are sufficiently large, a F reiman isomorphism of order ℓ allo ws us to transfer the integer b ound to Z k . T o a v oid assumptions on the prime factors of k , one may alternativ ely use Lev’s results. Our main contribution is con tained in Section 3, where we obtain non-trivial anticoncen tration b ounds for every ℓ ≥ 3. More precisely , if p > 2 λ  ℓ 0 3  ν (where ℓ 0 is a pow er of three such that ℓ 0 ≤ ℓ ), and if Y = Y 1 + · · · + Y ℓ with Y i i.i.d. and distributions uniformly b ounded by λ ≤ 9 / 10 (this restriction is only needed to provide an explicit v alue of ν ), then there exists an absolute constan t ν > 0 such that max x ∈ Z p P [ Y = x ] ≤ λ  3 ℓ 0  ν . In summary , while Lev’s asymptotic metho ds dominate in the large- ℓ regime, our approach pro vides explicit and self-contained bounds for small and mo derate v alues of ℓ , and applies also to distributions that are not necessarily uniform, where asymptotic estimates are not yet effective. 2. Known inequalities In this section, given a set A = { v 1 , v 2 , . . . , v n } of distinct in tegers, w e c onsider the random v ariable Y given by the sum of ℓ indep enden t v ariables Y 1 , . . . , Y ℓ that are uniformly distributed on A . W e establish an upp er b ound on P [ Y = x ] corresp onding to a special case of Theorem 2.3 in [13]. Although this follows from their more general result, we provide a direct and self-contained pro of, whic h we b eliev e ma y b e of indep endent in terest. First of all w e restrict ourself to consider A = {− ( n − 1) / 2 , . . . , ( n − 1) / 2 } , indeed Theorem 2 of [14] state that: Theorem 2.1 (Leader and Radcliffe) . Denote d by ˜ Y the r andom variable given by the sum of ℓ indep endent and uniformly distribute d on {− ( n − 1) / 2 , . . . , ( n − 1) / 2 } variables ˜ Y 1 , . . . , ˜ Y ℓ and 3 c onsider e d Y define d as ab ove, we have that  max x ∈ Z P [ Y = x ]  ≤  max x ∈ Z P [ ˜ Y = x ]  = P [ ˜ Y = M ] wher e M = ( 0 if ℓ ≡ 0 (mod 2) or n ≡ 1 (mo d 2); − 1 2 otherwise . Then our main to ol to upp er-bound P [ ˜ Y = x ] is the Berry-Esseen theorem which sa ys that: Theorem 2.2 (Berry-Esseen) . L et Y 1 , . . . , Y ℓ ′ b e indep endent r andom variables with the same dis- tribution and such that: (a) E ( Y j ) = 0 ; (b) E ( Y 2 j ) = σ 2 > 0 ; (c) E ( | Y j | 3 ) < ∞ . Then, set ρ = E ( | Y j | 3 ) σ 3 we have sup x       P [ 1 σ √ ℓ ′ ℓ ′ X j =1 Y j ≤ x ] − Φ( x )       ≤ C ρ √ ℓ ′ wher e Φ( x ) is the cumulative distribution function of the standar d normal distribution and C ≤ 0 . 4748 (se e [22]) is an absolute c onstant. Then we can state the following: Theorem 2.3. L et A = { v 1 , v 2 , . . . , v n } b e distinct inte gers and let Y b e the sum of ℓ indep endent and uniformly distribute d on A variables Y 1 , . . . , Y ℓ . Then, ther e exists an absolute c onstant D for which  max x ∈ Z P [ Y = x ]  ≤ D n √ ℓ − 1 . In p articular, however we take ϵ , if ℓ is sufficiently lar ge with r esp e ct to ϵ we have that  max x ∈ Z P [ Y = x ]  ≤ ϵ n . Pr o of. Due to Theorem 2.1, we can assume A = {− ( n − 1) / 2 , . . . , ( n − 1) / 2 } . W e note that, due to the indep endence of the Y i , P [ Y = x ] = (1) X y ∈ [ x − n − 1 2 ,x + n − 1 2 ] P [ Y 1 + · · · + Y ℓ − 1 = y ] P [ Y ℓ = x − y ] = P [ Y 1 + · · · + Y ℓ − 1 ∈ [ x − n − 1 2 , x + n − 1 2 ]] n . No w w e w an t to estimate P [ Y 1 + · · · + Y ℓ − 1 ∈ [ x − n − 1 2 , x + n − 1 2 ]] through Theorem 2.2. Indeed, set Ψ( z ) = P [ 1 σ √ ℓ − 1 ℓ − 1 X j =1 Y j ≤ z ] and set (Ψ − Φ)( z ) := Ψ( z ) − Φ( z ), w e hav e: P [ Y 1 + · · · + Y ℓ − 1 ∈ [ x − n − 1 2 , x + n − 1 2 ]] = Ψ ( x + n − 1 2 ) σ √ ℓ − 1 ! − Ψ ( x − n − 3 2 ) σ √ ℓ − 1 ! (2) = (Ψ − Φ) ( x + n − 1 2 ) σ √ ℓ − 1 ! − (Ψ − Φ) ( x − n − 3 2 ) σ √ ℓ − 1 ! + 4 Φ ( x + n − 1 2 ) σ √ ℓ − 1 ! − Φ ( x − n − 3 2 ) σ √ ℓ − 1 ! . This implies that, due to the triangular inequality , P [ Y 1 + · · · + Y ℓ − 1 ∈ [ x − n − 1 2 , x + n − 1 2 ]] ≤ (3)      (Ψ − Φ) ( x + n − 1 2 ) σ √ ℓ − 1 !      +      (Ψ − Φ) ( x − n − 3 2 ) σ √ ℓ − 1 !      +      Φ ( x + n − 1 2 ) σ √ ℓ − 1 ! − Φ ( x − n − 3 2 ) σ √ ℓ − 1 !      . W e note that the first t w o terms of Equation (3) can be upp er b ounded via the Berry-Esseen theorem. Here we hav e that σ = n 2 √ 3 (1 + o (1)) and ρ = 3 √ 3 4 (1 + o (1)) . Therefore (4)      (Ψ − Φ) ( x + n − 1 2 ) σ √ ℓ − 1 !      +      (Ψ − Φ) ( x − n − 3 2 ) σ √ ℓ − 1 !      ≤ 2 C 3 √ 3 4 √ ℓ − 1 (1 + o (1)) . F or the latter term, we note that      Φ ( x + n − 1 2 ) σ √ ℓ − 1 ! − Φ ( x − n − 3 2 ) σ √ ℓ − 1 !      ≤ max x | ϕ ( x ) | n σ √ ℓ − 1 . Hence, since the density function ϕ ( · ) of the standard normal distribution has a maxim um 1 2 π in 0, w e hav e (5)      Φ ( x + n − 1 2 ) σ √ ℓ − 1 ! − Φ ( x − n − 3 2 ) σ √ ℓ − 1 !      ≤ 1 2 π 2 √ 3 √ l − 1 (1 + o (1)) . Summing up Equations (4) and (5), w e hav e that P [ Y 1 + · · · + Y ℓ − 1 ∈ [ x − n − 1 2 , x + n − 1 2 ]] ≤ D √ ℓ − 1 where D is an absolute constant that also incorp orates the term o (1) whic h is b ounded and go es to zero when n go es to infinite. Placing this b ound in Equation (1), we obtain that P [ Y = x ] ≤ D n √ ℓ − 1 whic h implies the thesis. □ Ev en though for ℓ = 3 the previous theorem provides a trivial b ound, we can deal with this case directly . More precisely , we obtain the following non-trivial b ound: Theorem 2.4. L et A = { v 1 , v 2 , . . . , v n } b e distinct inte gers, and let Y 1 , Y 2 , Y 3 b e indep endent and uniformly distribute d on A . Then, set Y = Y 1 + Y 2 + Y 3 , we have that  max x ∈ Z P [ Y = x ]  ≤ 3 + 1 /n 2 4 n . 5 Pr o of. Due to Theorem 2.1, w e can assume A = { 1 , . . . , n } (which is a translate of {− ( n − 1) / 2 , . . . , ( n − 1) / 2 } ). Also, Theorem 2.1 sa ys that P [ Y = x ] is maximal for x = M = ⌊ 3( n +1) 2 ⌋ when A = { 1 , . . . , n } (which corresp onds to x = 0 or − 1 2 for {− ( n − 1) / 2 , . . . , ( n − 1) / 2 } ). Now we compute the n um b er of triples T = { ( y 1 , y 2 , y 3 ) ∈ A 3 : y 1 + y 2 + y 3 = M } . W e consider explicitly only the case n odd: the case n ev en is completely analogous. Clearly y 1 + y 2 + y 3 = M implies that y 1 + y 2 ∈ [ M − n, M − 1] = [ n +3 2 , 3 n +1 2 ] and that y 3 = M − y 2 − y 1 . Giv en x ∈ [ n +3 2 , 3 n +1 2 ], w e need to find the num ber of pair ( y 1 , y 2 ) which sum to x . Here w e hav e that, for any y 2 suc h that 1 ≤ x − y 2 ≤ n , there exists exactly one y 1 suc h that y 1 + y 2 = x . It follo ws that, if x ≤ n there are exactly x − 1 pairs y 1 , y 2 whic h sums to x . Similarly , if x ≥ n + 1, there are exactly 2 n − ( x − 1) pairs y 1 , y 2 whic h sums to x . Noting that the num ber of pairs that sum to x is equal to that of the pairs that sum to 2 n + 2 − x , we obtain that: | T | = X x ∈ [ n +3 2 ,n ] ( x − 1) + X x ∈ [ n +1 , 3 n +1 2 ] (2 n − ( x − 1)) = (6) 2 X x ∈ [ n +3 2 ,n ] ( x − 1) + n = 2   n − 1 X x =1 x − n − 1 2 X x =1 x   + n = 3 n 2 + 1 4 . The thesis follo ws b ecause P [ Y = x ] = { ( y 1 , y 2 , y 3 ) ∈ A 3 : y 1 + y 2 + y 3 = x } n 3 ≤ T n 3 ≤ 3 + 1 /n 2 4 n . □ 2.1. Kno wn inequalities for Z k and Z p . In the previous discussion, w e ha v e presen ted an upp er b ound on P [ Y = x ] for Z . It is then natural to inv estigate the case of Z k . Here, given a subset A of size n , if the prime factors of k are large enough, there exists a F reiman isomorphism of order ℓ from A to a subset B ⊆ Z . W e recall that (see T ao and V u, [23]). Definition 2.5. Let k ≥ 1, and let A , B b e additive sets with ambien t groups V and W resp ectively . A F reiman homomorphism of order ℓ , sa y ϕ , from ( A, V ) to ( B , W ) (or more succinctly from A to B ) is a map ϕ : A → B with the prop ert y that: a 1 + a 2 + · · · + a ℓ = a ′ 1 + a ′ 2 + · · · + a ′ ℓ → ϕ ( a 1 ) + ϕ ( a 2 ) + · · · + ϕ ( a ℓ ) = ϕ ( a ′ 1 ) + ϕ ( a ′ 2 ) + · · · + ϕ ( a ′ ℓ ) . If in addition there is an inv erse map ϕ − 1 : B → A which is also a F reiman homomorphism of order ℓ from ( B , W ) to ( A, V ), then we say that ϕ is a F reiman isomorphism of order ℓ and that ( A, V ) and ( B , W ) are F reiman isomorphic of order ℓ . Also from the result of Lev [17] (see also [5] and Theorem 3.1 of [6] where similar b ounds w ere giv en), we ha v e that Theorem 2.6 (Lev, [17]) . L et A b e a subset of size n of Z k wher e the prime factors of k ar e lar ger than n n . Then ther e exists a F r eiman isomorphism of any or der ℓ ≤ n , fr om A to a subset B ⊆ Z . Therefore, the result ov er the integers, namely Theorem 2.3, implies that Theorem 2.7. L et A b e a subset of size n of Z k wher e the prime factors of k ar e lar ger than n n . Then, set Y = Y 1 + · · · + Y ℓ the sum of ℓ indep endent uniformly distribute d (over the same set A ) r andom variables Y i , P [ Y = x ] ≤ D n √ ℓ − 1 wher e D is an absolute c onstant. 6 Note that this b ound has quite a strong hypothesis on the prime factors of k . Another, quite general, inequality can b e derived from the results of Lev ([15, 16]) who prov e that Theorem 2.8 (Lev, [15]) . The numb er of solutions of a 1 + a 2 + · · · + a ℓ = x , wher e x ∈ Z p , a i ∈ A ⊆ Z p , and | A | = n is at most N λ ( A ℓ ) ≤ 1 p n ℓ + r 8 π ℓ n ℓ − 1  1 + 2 ℓ − 1 / 2 + (3 / 4) ℓ/ 2+3 ℓ 3 / 2  . F rom this theorem, it immediately follo ws that Corollary 2.9. L et A b e a subset of size n of Z k . Then, set Y = Y 1 + · · · + Y ℓ the sum of ℓ indep endent uniformly distribute d (over the same set A ) r andom variables Y i , P [ Y = x ] ≤ N λ ( A ℓ ) n ℓ and henc e P [ Y = x ] ≤ 1 p + 1 n r 8 π ℓ  1 + 2 ℓ − 1 / 2 + (3 / 4) ℓ/ 2+3 ℓ 3 / 2  . No w, set ˜ C := q 8 π ℓ  1 + 2 ℓ − 1 / 2 + (3 / 4) ℓ/ 2+3 ℓ 3 / 2  , the b ound of Corollary 2.9 can b e written as P [ Y = x ] ≤ 1 p + ˜ C n whic h has, for p and ℓ large enough, the same b eha vior ˜ D n √ ℓ of the b ound of Theorem 2.3. 3. Anticoncentra tion for small ℓ In the previous Section, w e hav e review ed the kno wn anticoncen tration inequalities on P [ Y = x ]. In particular, in Z p , the b ound of Corollary 2.9 app ears to b e very p o w erful for large v alues of ℓ . On the other hand, a direct computation shows that, if ℓ is small (i.e., ℓ < 24), the constan t ˜ C b ecomes larger than one, and that bound b ecomes trivial. This section aims to pro vide a non trivial b ound also in this range. No w we consider a set A = { v 1 , v 2 , . . . , v n } of distinct elemen ts of Z p , and the random v ariable Y giv en by the sum of ℓ indep enden t and uniformly distributed on A v ariables Y 1 , . . . , Y ℓ . The goal is here to pro vide an upp erbound on  max x ∈ Z p P [ Y = x ]  . First of all, we consider the case ℓ = 3, and we pro v e the follo wing Lemma. Lemma 3.1. L et n ≥ 2 , p > 2 n , A = { v 1 , v 2 , . . . , v n } b e a set of distinct elements of Z p such that − v ∈ A whenever v ∈ A and let Y 1 , Y 2 and Y 3 b e indep endent variables which ar e uniformly distribute d on A . Then ther e exists an absolute c onstant C 1 < 1 1 , such that, set Y = Y 1 + Y 2 + Y 3 , we have:  max x ∈ Z p P [ Y = x ]  ≤ C 1 n . Pr o of. First of all, w e note that, if n = 2, w e may supp ose, without loss of generalit y , that A = {− 1 , 1 } . F or this set, with the same pro of of Theorem 2.4, we obtain that P [ Y = x ] ≤ 3+1 / 4 4 n (whic h giv e a b etter constan t than C 1 n ). So, in the following, we only consider the case n ≥ 3. F ollowing the pro of of Proposition 6.1 of [8], set f = P n i =1 1 n δ v i , we hav e that its discrete F ourier transform is ˆ f ( k ) = n X i =1 1 n e − 2 π iv i k/p . 1 Here the b est approximation we hav e for this constant is C 1 < 0 . 99993 . 7 Then, the probabilit y that Y 1 + Y 2 = x can b e written as P [ Y 1 + Y 2 = x ] = ( f ∗ f )( x ) = 1 p p − 1 X k =0 e 2 π ixk/p ( ˆ f ( k )) 2 = 1 p p − 1 X k =0 e 2 π ixk/p n X i =1 1 n e − 2 π iv i k/p ! 2 . Here we note that, since the set A is symmetric with resp ect to 0, P [ Y 1 + Y 2 = x ] = P [ Y 1 + Y 2 = − x ] . Th us we can write (7) P [ Y 1 + Y 2 = x ] = 1 p p − 1 X k =0 cos(2 π xk /p ) n X i =1 1 n e − 2 π iv i k/p ! 2 . Since, due again to the symmetry of A , we also hav e that b oth e − 2 π iv i k/p and e 2 π iv i k/p app ears in ˆ f ( k ), Equation (7) can b e written as: (8) P [ Y 1 + Y 2 = x ] = 1 p p − 1 X k =0 cos(2 π xk /p ) n X i =1 1 n cos(2 π v i k /p ) ! 2 . No w we consider B ⊆ Z p of cardinality n such that P [ Y 1 + Y 2 ∈ B ] is maximal and t w o p ositiv e real num bers ϵ 1 , ϵ 2 ≤ 1 / 24. W e divide B in to t w o sets defined as follo ws: B 1 := { x ∈ B : P [ Y 1 + Y 2 = x ] ≥ 1 − ϵ 1 n } ; B 2 := B \ B 1 = { x ∈ B : P [ Y 1 + Y 2 = x ] < 1 − ϵ 1 n } . No w we divide the pro of in to tw o cases. CASE 1: | B 1 | ≥ (1 − ϵ 2 ) n . Here we note that (9) P [ Y 1 + Y 2 = 0] = 1 n = 1 p p − 1 X k =0 ( ˆ f ( k )) 2 . This means that, for the v alues x ∈ B 1 , we can not lose to o muc h by inserting the cosine. Indeed, for these v alues, w e hav e that (10) P [ Y 1 + Y 2 = x ] = 1 p p − 1 X k =0 cos(2 π xk /p )( ˆ f ( k )) 2 ≥ 1 − ϵ 1 n . T aking the av erage o v er the v alues x ∈ B 1 , we obtain 1 | B 1 | P [ Y 1 + Y 2 ∈ B 1 ] = 1 | B 1 | X x ∈ B 1 1 p p − 1 X k =0 cos(2 π xk /p )( ˆ f ( k )) 2 = (11) = 1 p p − 1 X k =0  ( ˆ f ( k )) 2 P x ∈ B 1 cos(2 π xk /p ) | B 1 |  = 1 p + 1 p p − 1 X k =1  ( ˆ f ( k )) 2 P x ∈ B 1 cos(2 π xk /p ) | B 1 |  , 8 where the last equalities hold b y commuting the sums and noting that ˆ f (0) = 1. Now, Equations (9) and (10) implies that 1 | B 1 | P [ Y 1 + Y 2 ∈ B 1 ] ≥ 1 − ϵ 1 n = 1 p p − 1 X k =0  ( ˆ f ( k )) 2 (1 − ϵ 1 )  = = 1 − ϵ 1 p + 1 p p − 1 X k =1  ( ˆ f ( k )) 2 (1 − ϵ 1 )  . No w since 2 n < p , Equation (11) implies the existence of ¯ k  = 0 suc h that (12) X x ∈ B 1 cos(2 π x ¯ k /p ) | B 1 | ≥ (1 − 2 ϵ 1 ) . Indeed, otherwise, w e would ha v e that 1 p + 1 p p − 1 X k =1  ( ˆ f ( k )) 2 (1 − 2 ϵ 1 )  > 1 p + 1 p p − 1 X k =1  ( ˆ f ( k )) 2 P x ∈ B 1 cos(2 π xk /p ) | B 1 |  ≥ 1 − ϵ 1 p + 1 p p − 1 X k =1  ( ˆ f ( k )) 2 (1 − ϵ 1 )  that is, since 2 n ≤ p , ϵ 1 p > 1 p p − 1 X k =1 ( ˆ f ( k )) 2 ϵ 1 = ϵ 1  1 n − 1 p  ≥ ϵ 1 p whic h is a con tradiction. Also, note that since Z p is a field, we ma y suppose, without loss of generalit y , that ¯ k = 1. This means that (13) X x ∈ B 1 cos(2 π x/p ) | B 1 | ≥ (1 − 2 ϵ 1 ) . Hence the set B ′ 1 := { x ∈ B 1 : x/p ∈ [ − 1 / 6 , 1 / 6] } = { x ∈ B 1 : cos(2 π x/p ) ∈ [1 / 2 , 1] } has size larger than (1 − 2 ϵ 2 − 4 ϵ 1 ) n . Indeed, since | B | ≤ n , | B ′ 1 | < (1 − 2 ϵ 2 − 4 ϵ 1 ) n would imply that n + (1 − 2 ϵ 2 − 4 ϵ 1 ) n 2 ≥ | B 1 | + | B ′ 1 | 2 = | B ′ 1 | + | B 1 | − | B ′ 1 | 2 . Then we w ould obtain (1 − ϵ 2 − 2 ϵ 1 ) n > | B ′ 1 | + | B 1 \ B ′ 1 | 2 ≥ X x ∈ B 1 cos(2 π x/p ) ≥ (1 − 2 ϵ 1 )(1 − ϵ 2 ) n whic h is a contradiction. No w w e come back to the estimation of max x ∈ Z p P [ Y = x ]. Because of the indep endence of the v ariables Y 1 , Y 2 and Y 3 , we ha v e (14) P [ Y = x ] = X v i ∈ A P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] . No w we split this computation according to whether x − v i b elongs to B ′ 1 to B ′′ 1 := B 1 \ B ′ 1 or B 2 . (15) X v i ∈ A : x − v i ∈ B ′ 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] + X v i ∈ A : x − v i ∈ B ′′ 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ]+ 9 + X v i ∈ A : x − v i ∈ B 2 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] . Due to Equations (9) and (10), P [ Y 1 + Y 2 = x − v i ] is alw a ys b ounded by 1 /n . Then, since |{ v i ∈ A : x − v i ∈ B ′′ 1 }| ≤ (4 ϵ 1 + 2 ϵ 2 ) n , we can provide the following b ounds: X v i ∈ A : x − v i ∈ B ′ 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] ≤ 1 n |{ v i ∈ A : x − v i ∈ B ′ 1 }| n ; X v i ∈ A : x − v i ∈ B ′′ 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] ≤ 4 ϵ 1 + 2 ϵ 2 n ; and X v i ∈ A : x − v i ∈ B 2 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] < 1 − ϵ 1 n |{ v i ∈ A : x − v i ∈ B 2 }| n ≤ ϵ 2 (1 − ϵ 1 ) n . Therefore P [ Y = ¯ x ] ≥ 1 − ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n (the right-hand side of this inequalit y will b e denoted in the follo wing by C 1 n ) implies that 1 n |{ v i ∈ A : ¯ x − v i ∈ B ′ 1 }| n + 4 ϵ 1 + 2 ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n 2 ≥ 1 − ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n . As a consequence, we ha v e |{ v i ∈ A : ¯ x − v i ∈ B ′ 1 }| ≥ (1 − 4 ϵ 1 − 3 ϵ 2 ) n and hence, set A 1 := { v i ∈ A : v i ∈ [ − 1 / 6 − ¯ x, 1 / 6 − ¯ x ] } , w e hav e that (16) | A 1 | ≥ |{ v i ∈ A : ¯ x − v i ∈ B ′ 1 }| ≥ (1 − 4 ϵ 1 − 3 ϵ 2 ) n. On the other hand, given t w o triples v 1 , v 2 , v 3 and v ′ 1 , v ′ 2 , v ′ 3 ∈ A 1 , w e ha v e that v 1 + v 2 + v 3 = v ′ 1 + v ′ 2 + v ′ 3 if and only if they hav e the same sum also in Z (with the trivial identification). W e note that, since ϵ 1 , ϵ 2 ≤ 1 / 24 and n ≥ 3, n (1 − 4 ϵ 1 − 3 ϵ 2 ) > 2. Hence, due to Theorem 2.4, max x ∈ Z p P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] ≤ 3 + 1 / 9 4 n (1 − 4 ϵ 1 − 3 ϵ 2 ) . Here we ha v e that P [ Y = x | Y 1 ∈ A 1 ] = P [ Y = x | Y 2 ∈ A 1 ] = P [ Y = x | Y 3 ∈ A 1 ] , and that P [ Y = x | Y 1 ∈ A 1 ] can b e written as X y ∈ Z p P [ Y 1 + Y 2 = y | Y 1 ∈ A 1 ] P [ Y 3 = x − y ] ≤ X y ∈ Z p P [ Y 1 + Y 2 = y | Y 1 ∈ A 1 ] 1 n = 1 n . Hence we can b ound P [ Y = x ] as follows: P [ Y = x ] ≤ P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] + 3 P [ Y = x | Y 1 ∈ A 1 ] P [ Y 1 ∈ A 1 ] ≤ (17) 3 + 1 / 9 4 n (1 − 4 ϵ 1 − 3 ϵ 2 ) + 3 n | A \ A 1 | n ≤ 3 + 1 / 9 4 n (1 − 4 ϵ 1 − 3 ϵ 2 ) + 3 4 ϵ 1 + 3 ϵ 2 n . No w we consider the pair ( ϵ 1 , ϵ 2 ) to satisfy 3 + 1 / 9 4 n (1 − 4 ϵ 1 − 3 ϵ 2 ) + 3 4 ϵ 1 + 3 ϵ 2 n ≤ 1 − ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n . 10 Note that the set of these pairs is clearly non-empty , since (0 , 0) satisfies this relation with the strong inequality . Hence, there is a nontrivial set of p ositiv e pairs ϵ 1 , ϵ 2 ≤ 1 / 24 whic h satisfy it. F or these pairs, w e obtain that P [ Y = x ] ≤ 1 − ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n = C 1 n and w e conclude CASE 1 b y computing (with Mathematica) the minim um p ossible C 1 in this range, whic h is C 1 < 0 . 99993. CASE 2: | B 1 | < (1 − ϵ 2 ) n . Here we note that Equation (14) can b e split as follows: P [ Y = x ] = (18) X v i ∈ A : x − v i ∈ B 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] + X v i ∈ A : x − v i ∈ B 2 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] . No w we can provide the following b ounds: X v i ∈ A : x − v i ∈ B 1 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] ≤ |{ v i ∈ A : x − v i ∈ B 1 }| n 1 n and X v i ∈ A : x − v i ∈ B 2 P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] ≤ |{ v i ∈ A : x − v i ∈ B 2 }| (1 − ϵ 1 ) n 1 n . It follows that (19) P [ Y = x ] ≤ |{ v i ∈ A : x − v i ∈ B 1 }| n 1 n + |{ v i ∈ A : x − v i ∈ B 2 }| (1 − ϵ 1 ) n 1 n . W e note that the right-hand side of Equation (23) increases with | B | . Therefore, since | B 1 | < (1 − ϵ 2 ) n , we upp er-b ound the right hand side of Equation (23) by assuming that |{ v i ∈ A : x − v i ∈ B 1 }| = (1 − ϵ 2 ) n and |{ v i ∈ A : x − v i ∈ B 2 }| = ϵ 2 n. Summing up, we obtain that (20) P [ Y = x ] ≤ 1 − ϵ 2 n + ϵ 2 (1 − ϵ 1 ) n = C 1 n whic h concludes CASE 2. □ Prop osition 3.2. L et n ≥ 2 , p > 2 n , A = { v 1 , v 2 , . . . , v n } b e distinct elements of Z p and let Y 1 , Y 2 and Y 3 b e indep endent variables which ar e uniformly distribute d on A . Then ther e exists C 2 < 1 2 , such that, set Y = Y 1 + Y 2 + Y 3 , we have:  max x ∈ Z p P [ Y = x ]  ≤ C 2 n . Pr o of. First of all, w e note that, if n = 2, w e may supp ose, without loss of generalit y , that A = {− 1 , 1 } . F or this set, with the same pro of of Theorem 2.4, we obtain that P [ Y = x ] ≤ 3+1 / 4 4 n (whic h giv e a b etter constan t than C 2 n ). So, in the following, we only consider the case n ≥ 3. Let ¯ x b e suc h that P [ Y 1 + Y 2 = ¯ x ] is maximal. Since Z p is a field, w e may suppose, without loss of generality , that ¯ x = 0. Set ϵ 3 suc h that C 1 (1 − ϵ 3 ) + 3 ϵ 3 = 1 − ϵ 3 and C 2 = 1 − ϵ 3 . W e note that this relation has a solution ϵ 3 ∈ (0 , 1 − C 1 ): indeed, in 0 its left-hand side is smaller than its right-hand side, while in 1 − C 1 the opp osite inequality holds. Also, we can compute, computationally , the v alues of ϵ 3 and C 2 , and note that C 2 < 0 . 999986. Then we divide the pro of into t w o cases. 2 Here the b est approximation we hav e for this constant is C 2 < 0 . 999986 . 11 CASE 1: P [ Y 1 + Y 2 = 0] ≥ 1 − ϵ 3 n . Here we define A 1 := { x ∈ A : − x ∈ A } . W e ha v e that P [ Y 1 + Y 2 = 0] = P [ Y 1 ∈ A 1 ] · P [ Y 2 = − Y 1 ] = | A 1 | n 2 . This implies that (21) | A 1 | ≥ n (1 − ϵ 3 ) . Note that, since n ≥ 3, | A 1 | ≥ 2. Hence, due to Lemma 3.1, there exists C 1 suc h that P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] ≤ C 1 n (1 − ϵ 3 ) . Here we ha v e that P [ Y = x | Y 1 ∈ A 1 ] = P [ Y = x | Y 2 ∈ A 1 ] = P [ Y = x | Y 3 ∈ A 1 ] , and that P [ Y = x | Y 1 ∈ A 1 ] can b e written as X y ∈ Z p P [ Y 1 + Y 2 = y | Y 1 ∈ A 1 ] P [ Y 3 = x − y ] ≤ X y ∈ Z p P [ Y 1 + Y 2 = y | Y 1 ∈ A 1 ] 1 n = 1 n . Therefore, we can b ound P [ Y = x ] as follows: P [ Y = x ] ≤ P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] + 3 P [ Y = x | Y 1 ∈ A 1 ] P [ Y 1 ∈ A 1 ] ≤ (22) C 1 n (1 − ϵ 3 ) + 3 n | A \ A 1 | n ≤ C 1 n (1 − ϵ 3 ) + 3 ϵ 3 n = C 2 n whic h concludes CASE 1. CASE 2: P [ Y 1 + Y 2 = 0] < 1 − ϵ 3 n . Here we hav e that (23) P [ Y = x ] = X v i ∈ A P [ Y 1 + Y 2 = x − v i ] P [ Y 3 = v i ] . Since P [ Y 1 + Y 2 = x − v i ] ≤ P [ Y 1 + Y 2 = 0] < 1 − ϵ 3 n , Equation (23) can b e written as (24) P [ Y = x ] < X v i ∈ A 1 − ϵ 3 n P [ Y 3 = v i ] = 1 − ϵ 3 n = C 2 n . □ This prop osition can b e generalized to: Theorem 3.3. L et λ ≤ 9 10 , p > 2 λ , Y 1 , Y 2 and Y 3 b e identic al and indep endent variables such that  max x ∈ Z p P [ Y 1 = x ]  ≤ λ. Then ther e exists C 3 < 1 3 , such that, set Y = Y 1 + Y 2 + Y 3 we have:  max x ∈ Z p P [ Y = x ]  ≤ C 3 λ. 3 Here the b est approximation we hav e for this constant is C 3 < 1 − 1 . 3 · 10 − 12 . 12 Pr o of. W e consider p ositiv e ϵ 4 and ϵ 5 whic h satisfy the relation: 1 − ϵ 4 ϵ 5 ≥ 3( ϵ 4 + ϵ 5 ) + C 2 (1 − ϵ 4 ) 3 (1 − ϵ 5 ) and we set C 3 := 1 − ϵ 4 ϵ 5 . W e will also assume that ϵ 5 < 1 / 10 and ϵ 4 < 1. It is clear that, since (0 , 0) satisfies the ab o v e relation (with a strong inequalit y), the set of pairs sub ject to these constrain ts is nonempt y . Then, we define A 1 := { x ∈ Z p : P [ Y 1 = x ] ≥ λ (1 − ϵ 4 ) } . Then we divide the pro of in to tw o cases according to the cardinalit y of A 1 . CASE 1: | A 1 | ≥ (1 − ϵ 5 ) λ . W e wan t to pro vide an upp er b ound to P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] b y applying Prop osition 3.2. F or this purp ose, w e note that: P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] = (25) X y 1 ,y 2 ∈ A 1 P [ Y 1 = y 1 | Y 1 ∈ A 1 ] P [ Y 2 = y 2 | Y 2 ∈ A 1 ] P [ Y 3 = x − y 1 − y 2 | Y 3 ∈ A 1 ] . Here, for y 1 ∈ A 1 , we ha v e P [ Y 1 = y 1 | Y 1 ∈ A 1 ] P [ Y 1 ∈ A 1 ] = P [ Y 1 = y 1 ] ≤ λ. Since P [ Y 1 ∈ A 1 ] = X x ∈ A 1 P [ Y 1 = x ] ≥ | A 1 | λ (1 − ϵ 4 ) , it follows that P [ Y 1 = y 1 | Y 1 ∈ A 1 ] ≤ λ P [ Y 1 ∈ A 1 ] ≤ 1 | A 1 | (1 − ϵ 4 ) . Note that, named b y ˜ Y = ˜ Y 1 + ˜ Y 2 + ˜ Y 3 where ˜ Y 1 , ˜ Y 2 , ˜ Y 3 are uniform distribution o v er A 1 , we hav e that (26) P [ Y 1 = y 1 | Y 1 ∈ A 1 ] ≤ 1 | A 1 | (1 − ϵ 4 ) = 1 (1 − ϵ 4 ) P [ ˜ Y 1 = y 1 ] . Noting that P [ ˜ Y 1 = y 1 ] ≤ λ (1 − ϵ 5 ) , Equations (25) and (26) imply that (27) P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] ≤  1 1 − ϵ 4  3 P [ ˜ Y = x ] ≤  1 1 − ϵ 4  3 C 2 λ (1 − ϵ 5 ) . Here the last inequalit y holds b ecause, since λ ≤ 9 / 10, ϵ 5 < 1 / 10, and we are in CASE 1, we ha v e | A 1 | > 1. Moreov er, since | A 1 | is integer, this means that | A 1 | ≥ 2 and w e can apply Prop osition 3.2 to the distribution ˜ Y . Recalling that P [ Y 1 ∈ A 1 ] ≥ | A 1 | λ (1 − ϵ 4 ) ≥ (1 − ϵ 5 )(1 − ϵ 4 ) > (1 − ϵ 4 − ϵ 5 ) , w e hav e P [ Y 1 ∈ A 1 ] ≤ ϵ 4 + ϵ 5 . Pro ceeding as in Equation (22), it follows that P [ Y = x ] ≤ P [ Y = x | Y 1 , Y 2 , Y 3 ∈ A 1 ] + 3 P [ Y = x | Y 1 ∈ A 1 ] P [ Y 1 ∈ A 1 ] ≤ (28)  1 1 − ϵ 4  3 C 2 λ (1 − ϵ 5 ) + 3 λ P [ Y 1 ∈ A 1 ] ≤  1 1 − ϵ 4  3 C 2 λ (1 − ϵ 5 ) + 3 λ ( ϵ 4 + ϵ 5 ) ≤ C 3 λ whic h concludes CASE 1. CASE 2: | A 1 | < (1 − ϵ 5 ) λ . Here w e prov e that P [ Y 1 + Y 2 = x ] ≤ C 3 λ. Indeed, w e hav e that P [ Y 1 + Y 2 = x ] = X y 1 ∈ A 1 P [ Y 1 = y 1 ] P [ Y 2 = x − y 1 ] + X y 1 ∈ A 1 P [ Y 1 = y 1 ] P [ Y 2 = x − y 1 ] . 13 Here we note that (29) P [ Y 1 = y 1 ] ≤ ( λ if y 1 ∈ A 1 ; λ (1 − ϵ 4 ) if y 1 ∈ A 1 . Therefore P [ Y 1 + Y 2 = x ] ≤ λ   X y 1 ∈ A 1 P [ Y 2 = x − y 1 ] + (1 − ϵ 4 ) X y 1 ∈ A 1 P [ Y 2 = x − y 1 ]   = (30) = λ − λϵ 4 X y 1 ∈ A 1 P [ Y 2 = x − y 1 ] . No w we note that, named A 2 := { y 1 ∈ Z p : y 1 ∈ A 1 } , | A 2 | ≥ p − (1 − ϵ 5 ) λ and we ha v e that P [ Y 2 ∈ A 2 ] = 1 − P [ Y 2 ∈ A 1 ] ≥ 1 − λ | A 1 | ≥ ϵ 5 . Also, given another set B of the same cardinalit y , we ha v e that P [ Y 2 ∈ B ] ≥ P [ Y 2 ∈ A 2 ] ≥ ϵ 5 . Therefore, considering that |{ z ∈ Z p : z = x − y 1 , y 1 ∈ A 1 }| = | A 2 | , Equation (30) can b e written as P [ Y 1 + Y 2 = x ] ≤ λ − λϵ 4 P [ Y 2 ∈ { z ∈ Z p : z = x − y 1 , y 1 ∈ A 1 } ] ≤ λ − λϵ 4 ϵ 5 = C 3 λ. No w it is enough to note that, since Y 1 , Y 2 and Y 3 are indep enden t, P [ Y 1 + Y 2 + Y 3 = x ] = X y 3 ∈ Z p P [ Y 3 = y 3 ] P [ Y 1 + Y 2 = x − y 3 ] ≤ X y 3 ∈ Z p P [ Y 3 = y 3 ] C 3 λ ≤ C 3 λ whic h concludes CASE 2. The thesis follo ws showing, with Mathematica, that we can choose C 3 < 1 − 2 . 27 · 10 − 12 . □ Remark 3.4. In The or em 3.3, we may also we aken the hyp othesis that λ < 9 / 10 and assume that λ < 1 . With the same pr o of, cho osing ϵ 5 < 1 − λ , we find the existenc e of a c onstant C 3 ,λ < 1 also for values of λ that ar e close to one. On the other hand, this c onstant c annot b e made explicit, and it dep ends on λ . So, to obtain an absolute c onstant, we pr efer to assume λ smal ler than a given value (i.e. 9 / 10 ). As a consequence, we can state the following result, whic h is analogous, for the case of Z p , to that of the previous section. Theorem 3.5. L et us c onsider λ ≤ 9 10 , ϵ > 0 , p > 2 λϵ and let Y = Y 1 + · · · + Y ℓ wher e Y i ar e i.i.d. whose distributions ar e upp er-b ounde d by λ . Then, if ℓ is sufficiently lar ge with r esp e ct to ϵ , we have that (31)  max x ∈ Z p P [ Y = x ]  ≤ ϵλ. In p articular, if k 0 is a p ositive inte ger such that ( C 3 ) k 0 < ϵ ≤ ( C 3 ) k 0 − 1 and ℓ 0 = 3 k 0 , Equation (31) holds for any ℓ ≥ ℓ 0 . Pr o of. First of all, w e prov e, b y induction on k , that, assuming p > 2 C k − 1 3 λ , (32)  max x ∈ Z P [ Y 1 + Y 2 + · · · + Y 3 k = x ]  ≤ C k 3 λ. BASE CASE. The case k = 1 follows from Theorem 3.3. 14 INDUCTIVE STEP . W e assume p > 2 C k 3 λ and that Equation (32) is true for k and we pro v e it for k + 1. At this purp ose we set ˜ Y 1 = Y 1 + Y 2 + · · · + Y 3 k and we note that, set ˜ λ = C k 3 λ for the inductiv e hypothesis, P [ ˜ Y 1 = x ] ≤ ˜ λ. Therefore, b ecause of Theorem 3.3, w e see that if ˜ Y 2 , ˜ Y 3 , are three identical copies of ˜ Y 1 and the set Y = ˜ Y 1 + ˜ Y 2 + ˜ Y 3 , P [ Y = x ] ≤ C 3 ˜ λ = C k +1 3 λ. The inductive claim follows since Y = ˜ Y 1 + ˜ Y 2 + ˜ Y 3 = Y 1 + Y 2 + · · · + Y 3 k +1 . No w we consider ℓ ≥ 3 k . A b ound for this case follows, since we ha v e P [ Y = x ] = X y 3 P [ Y 3 k +1 + Y 3 k +2 + · · · + Y ℓ = y ] P [ Y 1 + Y 2 + · · · + Y 3 k = x − y ] ≤ X y P [ Y 3 k +1 + Y 3 k +2 + · · · + Y ℓ = y ] C k 3 λ ≤ C k 3 λ. No w, define k 0 to b e a p ositiv e in teger such that ( C 3 ) k 0 < ϵ ≤ ( C 3 ) k 0 − 1 and let ℓ 0 = 3 k 0 . Since p > 2 ϵλ > 2 λC k 0 − 1 3 , Equation (31) holds for an y ℓ ≥ ℓ 0 and the main claim of the theorem follows. □ Remark 3.6. In The or em 3.5, we may also we aken the hyp othesis that λ < 9 / 10 and assume that λ < 1 . With the same pr o of, we find the existenc e of a c onstant C 3 ,λ < 1 for which the fol lowing statement holds. L et us c onsider ϵ > 0 , p > 2 λϵ and let Y = Y 1 + · · · + Y ℓ wher e Y i ar e i.i.d. whose distributions ar e upp er-b ounde d by λ . Then, if ℓ is sufficiently lar ge with r esp e ct to ϵ , we have that (33)  max x ∈ Z p P [ Y = x ]  ≤ ϵλ. In p articular, if k 0 is a p ositive inte ger such that ( C 3 ,λ ) k 0 < ϵ ≤ ( C 3 ,λ ) k 0 − 1 and ℓ 0 = 3 k 0 , Equation (33) holds for any ℓ ≥ ℓ 0 . 3.1. Conclusiv e Remarks. The imp ortance of this result is that it is non trivial also for small v alues of ℓ ≥ 3 provided that ϵ is sufficiently close to, but still b elo w, 1. W e recall that, on the other hand, the b ound of Corollary 2.9 is worse that 1 /n whenev er ℓ < 24. With our b ound (assuming p and λ satisfy the hypotheses of Theorem 3.5), if 3 ≤ ℓ < 9 we obtain that  max x ∈ Z p P [ Y = x ]  ≤ C 3 λ while, if 9 ≤ ℓ < 27 ,  max x ∈ Z p P [ Y = x ]  ≤ ( C 3 ) 2 λ. Moreo v er, set ν = − log 3 C 3 , Theorem 3.5 can b e restated as follo ws. Let us consider λ ≤ 9 10 , p > 2 λ  ℓ 0 3  ν , where ℓ 0 is a p o w er of three and ℓ ≥ ℓ 0 , and let Y = Y 1 + · · · + Y ℓ where Y i are i.i.d. whose distributions are upp er-b ounded b y λ . Then, the distribution Y is b ounded by λ  3 ℓ 0  ν , where ν is a p ositiv e absolute constan t. F rom this discussion, we can also estimate that ν > 1 . 19 · 10 − 12 . The v alue obtained here (that w e do not exp ect to b e tight) is a result of the univ ersal nature of our estimations whic h must hold for an y distribution b ounded by 9 / 10. 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