A Gale–Berlekamp permutation-switching problem in higher dimensions

European Journal of Combinatorics 77 (2019) 17–30

Contents lists available at ScienceDirect

European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc

A Gale–Berlekamp permutation-switching problem in higher dimensions Gustavo Araújo a , Daniel Pellegrino b a b

Departamento de Matemática, Universidade Estadual da Paraíba, 58.429-500 Campina Grande, Brazil Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900 João Pessoa, Brazil

article

info

Article history: Received 30 January 2018 Accepted 10 October 2018 Available online xxxx

a b s t r a c t ( )

Let an n × n array aij of lights be given, each either on (when aij = 1) or off (when aij = −1). For each row and each column there is a switch so that if the switch is pulled (xi = −1 for row i and yj = −1 for column j) all of the lights in that line are switched: on to off or off to on. The unbalancing lights problem (Gale–Berlekamp switching game) consists in maximizing the difference between the lights on and off. Fixed the scalar field K = R or C, we obtain the exact parameters for a generalization of the unbalancing lights problem in higher dimensions m ≥ 2, (which)consists in estimating,⏐ for any ⏐ given n × · · · × n array ai1 ···im of scalars satisfying ⏐ai1 ···im ⏐ = 1 and any given p ∈ [2, ∞], the expression

⏐ ⏐ ⏐ n ⏐ ∑ ⏐ ⏐ (1) (m) K ⏐ ⏐ gm (p) = max a x · · · x i1 ···im i1 ,n im ⏐ , ⏐ ⏐i1 ,...,im =1 ⏐ (j)

where the maximum is evaluated for all scalars xi

j

such that

(j) (xi )ni=1 p j

∥ = 1. In the particular case m = 2 and p = ∞, we ∥ recover the original problem. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction We begin by presenting a combinatorial game, sometimes called Gale–Berlekamp switching game or unbalancing lights problem (for a( presentation we refer, for instance, to the classical book of Alon ) and Spencer [2]). Let an n × n array aij of lights be given, each either on (when aij = 1) or off (when E-mail addresses: [email protected] (G. Araújo), [email protected] (D. Pellegrino). https://doi.org/10.1016/j.ejc.2018.10.007 0195-6698/© 2018 Elsevier Ltd. All rights reserved.

18

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

aij = −1). Let us also suppose that for each row and each column there is a switch so that if the switch is pulled (xi = −1 for row i and yj = −1 for column j) all of the lights in that line are switched: on to off or off to on. The problem consists in maximizing the difference between the lights on and off. A probabilistic approach (using the Central Limit Theorem) to this problem (see [2]) provides the following asymptotic estimate: Theorem 1.1 ([2, Theorem 2.5.1]). Let aij = ±1 for 1 ≤ i, j ≤ n. Then there exist xi , yj = ±1, 1 ≤ i, j ≤ n, such that n ∑

(√ aij xi yj ≥

2

π

i,j=1

) + o(1) n3/2 ,

(1)

and the exponent 3/2 is optimal. In other words, for any initial configuration aij (it is possible to) perform √ 2/π + o(1) n3/2 . switches so that the number of lights on minus the number of lights off is at least

( )

We recall that above, and from now on, the notation o(1) denotes a function on the variable n such that limn→∞ o(1) = 0. In higher dimensions (cf. mathoverflow.net/questions/59463/unbalancinglights-in-higher-dimensions, by A. Montanaro) the unbalancing ) lights problem is stated as follows: ( Let m ≥ 2 be an integer and let an n × · · · × n array ai1 ···im of lights be given each either on (when ai1 ···im = 1) or off (when ai1 ···im = −1). Let us also suppose that for each ij there is a switch so that if the switch is pulled (xij = −1) all of the lights in that line are ‘‘switched": on to off or off to on. The goal is to maximize the difference between the lights on and off. It is a well-known consequence of the Bohnenblust–Hille inequality [10] (see [12] for a modern (j) survey on this topic) that there exist xi = ±1, with 1 ≤ ij ≤ n and 1 ≤ j ≤ m, and a constant C ≥ 1, j such that n ∑

(1)

1

(m)

ai1 ···im xi · · · xim ≥ 1

i1 ,...,im =1

Cm

n

m+1 2

,

and the exponent (m + 1)/2 is sharp (see [17]). A step further suggested by A. Montanaro is to investigate if the term C m can be improved. Using recent estimates of the Bohnenblust–Hille inequality (j) (see [8, Corollary 3.3]) it is plain that there exist xi = ±1, with 1 ≤ i ≤ n and 1 ≤ j ≤ m such that n ∑

(1) 1

i1 ,...,im =1

1

(m)

ai1 ···im xi · · · xim ≥

1.3m0.365

n

m+1 2

,

(2)

and the exponent (m + 1)/2 is sharp. It is still an open problem if the term 1.3m0.365 (here and henceforth 1.3m0.365 is just a simplification of the estimate κ m(2−log 2−γ )/2 provided in [8, Corollary 3.3], where γ is the Euler–Mascheroni constant and κ is a certain constant smaller than 1.3) can be improved to a universal constant. As a matter of fact, when we deal with the complex scalar field, according to [8, Corollary 3.2] we can choose slightly better constants (of the form κ m0.212 ) but, for the sake of simplicity we shall use 1.3m0.365 which are valid for both cases. From now on K denotes the real scalar field R or the complex scalar field C. Also, following the usual notation, we denote

( n )1   ∑ ⏐⏐ (j) ⏐⏐p p  (j) n  (xi )i=1  := ⏐xi ⏐ p

(3)

i=1

for p < ∞, and

   (j) n  (xi )i=1 

∞

{⏐ ⏐ } ⏐ ⏐ := max ⏐xi(j) ⏐ : i = 1, . . . , n .

(4)

Moreover, ℓnp denotes Kn endowed with the norm (3) for p < ∞ or (4) for p = ∞; also, as usual, ℓp denotes the Banach space of all p-summable sequences of scalars endowed ( )∞with the respective norm (3) or (4), with n = ∞. Finally, c0 denotes the space of scalar sequences xj j=1 such that limn→∞ xn =

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

19

0, endowed with the sup norm. Finally, if E1 , . . . , Em are Banach spaces and T : E1 × · · · × Em → K is an m-linear form, we recall that

  { } ∥T ∥ := sup |T (x1 , . . . , xm )| : xj  ≤ 1 for all j = 1, . . . , m . It is also worth recalling the following well-known consequence of the Minkowski/Krein–Milman Theorem, that will be useful along this paper: Lemma 1.2 (Corollary of the Minkowski/Krein–Milman Theorem). For all m-linear forms T : ℓn∞ × · · · × ℓn∞ → R we have

⏐ ( )⏐ ∥T ∥ = max ⏐T x(1) , . . . , x(m) ⏐ , where x(j) has all entries equal to 1 or −1, for all j = 1, . . . , m. Some variants of the unbalancing lights problem have been already investigated (see [11,15] for interesting variations of the game). In this paper we consider a new variant:

(

)

Problem 1.3. Let⏐ m ≥ ⏐2 be an integer and let ai1 ···im be an n × · · · × n array of (real or complex) scalars such that ⏐ai1 ···im ⏐ = 1. For p ∈ (1, ∞], calculate

⏐ ⏐ ⏐ ⏐ ∑ n ⏐ ⏐ (1) (m) ⏐ K ⏐ ai1 ···im xi · · · xim ⏐ , gm,n (p) = max ⏐ 1 ⏐ ⏐i1 ,...,im =1 (j)

where the maximum is evaluated over all xi

n ∈ K such that ∥(x(j) i )i=1 ∥p = 1 for all j. When p = ∞

and K = R we recover the classical unbalancing lights problem in higher dimensions. In fact, in this (j) case Lemma 1.2 asserts that the maximum is achieved when |xi | = 1 for all i, j. (j)

We remark that, for p < ∞, the scalars xi considered in Problem 1.3 do not need to have modulus (j) one; they are arbitrary, and just need to satisfy ∥(xi )ni=1 ∥p = 1. The first main general result of this paper (Theorem 2.3), in particular, provides, as a straightforward consequence, the statement of the main technical result of this paper concerning the unbalancing lights problem (Corollary 2.5). It provides the sharp exponents for the unbalancing lights problem for p ≥ 2:

• If p ∈ [2, ∞] and m ≥ 2 is an integer, then mp+p−2m 1 n 2p 1.3m0.365 for all positive integers n. Moreover, the exponents (mp + p − 2m)/2p are sharp.

K gm ,n (p) ≥

(5)

Above and henceforth, when p = ∞, as usual in this field, the expression (mp + p − 2m)/2p means (m + 1)/2. More generally, if A ⊂ R and f : A → R is a function, we define f (∞) := limp→∞ (p) whenever it makes sense. 2. Results A first partial solution to Problem 1.3 is a straightforward consequence of the Hardy–Littlewood inequalities. The Hardy–Littlewood inequalities [14,16,21] for m-linear forms assert that for any K K integer m ≥ 2 there exist constants Cm ,p , Dm,p ≥ 1 such that

⏐⎛ ⎞ p−pm ⏐ n ⏐ ⏐ p ⏐⎝ ∑ ⏐⏐ ≤ DK T (ej1 , . . . , ejm )⏐ p−m ⎠ ⏐ m,p ∥T ∥ , if m < p ≤ 2m, ⏐ j ,..., j = 1 ⏐ 1 m ⏐⎛ p−2m ⎞ mp+2mp ⏐ n ⏐ ∑ 2mp ⏐ ⏐ ⏐⎝ ⏐T (ej , . . . , ejm )⏐ mp+p−2m ⎠ ≤ CmK,p ∥T ∥ , if p ≥ 2m, ⏐ 1 ⏐ j ,...,jm =1 1

(6)

20

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

for all m-linear forms T : ℓnp × · · · × ℓnp → K, and all positive integers n. Moreover, the exponents in both inequalities of (6) are sharp. K K The optimal constants Cm ,p , Dm,p are unknown; even the asymptotic behavior of these constants is K unknown. Up to now, the best estimates for Cm ,p can be found in [4,5]: K Cm ,p ≤

(√ ) 2m(mp −1) ( 2

1.3m0.365

) p−p2m

.

K 0.365 For p > 2m(m − 1)2 we also know from [4] that Cm ; it is not known if, in general, the ,p ≤ 1.3m K K same estimate is valid for the other choices of p. The notation of Cm ,p , Dm,p as the optimal constants of the Hardy–Littlewood inequalities will be kept all along the paper. By (6) we easily have the following:

Proposition 2.1. Let m ≥ 2 and n be positive integers and p ∈ (m, ∞]. Then

⏐ m(p−m) ⏐ K 1 ⏐gm,n (p) ≥ n p , for m < p ≤ 2m, K ⏐ D m,p ⏐ ⏐ mp+p−2m ⏐ K 1 ⏐gm,n (p) ≥ n 2p , for p ≥ 2m. ⏐ CK m,p

Among other results, our main result related to the unbalancing lights problem (Corollary 2.5) shows that the above estimates of Proposition 2.1 are far from being precise. We will show that:

• The exponent m (p − m) /p can be improved to (mp + p − 2m)/2p in the case m < p ≤ 2m; 0.365 • The constants CmK,p and DK ; m,p can be replaced by 1.3m • The inequalities from Proposition 2.1 are also valid for 2 ≤ p ≤ m with the constants 1.3m0.365 and exponents (mp + p − 2m)/2p; • All the above exponents (mp + p − 2m)/2p are optimal. Recently (see [1]), it has been shown that the constants DK m,p have essentially a very low growth but since we are now also improving the associated exponents, the prior estimates of DK m,p are not useful for our tasks. To achieve our goals, we begin by revisiting the Kahane–Salem–Zygmund inequality. It is a probabilistic result that furnishes unimodular multilinear forms (i.e. multilinear forms with coefficients with modulo 1) with ‘‘small’’ norms. This result is fundamental to the proof of the optimality of the exponents of the Hardy–Littlewood inequality. For p ≥ 1, the Kahane–Salem–Zygmund inequality asserts that there exists a m-linear form A : ℓnp × · · · × ℓnp −→ K of the form n ∑

A x(1) , . . . , x(m) =

(

)

δi1 ···im x(i11) · · · x(imm) ,

i1 ,...,im =1

with δi1 ···im ∈ {−1, 1}, such that 1

∥A∥ ≤ Cm n 2

( ) +m 12 − 1p

.

It is worth mentioning that in general the Kahane–Salem–Zygmund inequality is stated for complex scalars, but the case of real scalars is a well-known straightforward consequence of the complex case. Besides, for 1 ≤ p ≤ 2 a better estimate can essentially be found in [6]. So, we have the following: Theorem 2.2 (Kahane–Salem–Zygmund Inequality). Let n, m be positive integers and p ≥ 1. Then there exists a m-linear form A : ℓnp × · · · × ℓnp −→ K of the form A x(1) , . . . , x(m) =

(

)

n ∑ i1 ,...,im =1

δi1 ···im x(i11) · · · x(imm) ,

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

21

with δi1 ···im ∈ {−1, 1}, such that {

∥A∥ ≤ Cm n

)

(

max 21 +m 12 − 1p ,1− 1p

}

.

Before presenting our main result, let us introduce some required definitions for its proof. If E is a Banach space, the symbol BE ∗ will denote the closed unit ball of the topological (dual E. For s ≥ 1 ( )∞ ( of ))∞ we represent by ℓw s (E) the linear space of the sequences xj j=1 in E such that ϕ xj j=1 ∈ ℓs for every continuous linear functional ϕ : E → K. Let E1 , . . . , Em be Banach spaces and p, q ∈ [1, +∞). A multilinear form T : E1 × · · · × Em → K is multiple (q, p)-summing if there exists a constant C > 0 such that

⎛

⎞ 1q

∞ ∑

q

(m) ⎠ ≤ C ∥T ∥ |T (x(1) j1 , . . . , xjm )|

⎝ j1 ,...,jm =1

for all

(k) (xj )∞ j=1

m ∏ k=1

⎛ sup ⎝

∞ ∑

ϕk ∈BE ∗

⎞ 1p p

⎠ |ϕk (x(k) j )|

j=1

k

∈ ℓw p (Ek ). For more information and recent results on multiple summing operators we

refer to [7,20] and the references therein. Now we are able to state and prove the first main result of this paper. It is a kind of inequality of Hardy–Littlewood type for unimodular m-linear forms on ℓp -spaces and p > 1. Theorem 2.3. Let m, n be positive integers, p ∈ (1, ∞] and T : ℓnp × · · · × ℓnp → K be a unimodular m-linear form. Let r ∈ (0, ∞] (when r = ∞ we consider the sup norm in the left-hand-side of (7)) and Cm,p > 0 be such that

⎛

⎞ 1r ⏐r ⏐ ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ Cm,p ∥T ∥ . 1

n ∑

⎝

(7)

j1 ,...,jm =1

Then (1) If p ∈ [2, ∞], we have (7) if, and only if, r ≥ (2) If p ∈

,

, ],

( m2m 2) +1 (1 m2m +1

2mp we have (7) for r mp+p−2m mp the optimal r belongs to p−1

≥

2mp mp+p−2m

.

and the optimal r satisfies r ≥

mp . p−1

(3) If p ∈ , [ , ∞]. Moreover, in all cases in which we have proved that the inequality holds, we can choose Cm,p = 1.3m0.365 . Proof. Using the isometric characterization of the spaces of ℓw 1 (c0 ) (see [13, Proposition 2.2]), by the Bohnenblust–Hille inequality we know that every continuous m-linear form is multiple (2m/(m + 1), 1)-summing with constant dominated by 1.3m0.365 . Thus, denoting by p∗ the conjugate of p, i.e., p∗ = p/(p − 1) and p∗ = 1 when p = ∞, and considering p > 2m/(m + 1), we have

⎛

+1 ⎞ m2m ⎛ ⎞m n ∑ ⏐ ⏐ 2m ⏐ ⏐ ⏐ϕj ⏐⎠ ⏐T (ej , . . . , ejm )⏐ m+1 ⎠ ≤ 1.3m0.365 ∥T ∥ ⎝ sup 1

n ∑

⎝

ϕ∈Bℓn

j1 ,...,jm =1

p∗ j=1

for all m-linear forms T : ℓ × · · · × ℓ → K. Hence, considering unimodular m-linear forms, we have n p

(

n

m

+1 ) m2m

n p

( ≤ 1.3m

0.365

∥T ∥

n 1

)m ,

n p∗ and finally

∥T ∥ ≥

1 1.

3m0.365

n

mp+p−2m 2p

,

(8)

22

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

for all unimodular m-linear forms T : ℓnp × · · · × ℓnp → K. Note that here it is important to note that p>

2m m+1

⇒

mp + p − 2m 2p

> 0.

From (8), recalling that we are dealing with unimodular m-linear forms, we have

⎛

p−2m ⎞ mp+2mp ⏐ 2mp ⏐ ⏐T (ej , . . . , ejm )⏐ mp+p−2m ⎠ ≤ 1.3m0.365 ∥T ∥ 1

n ∑

⎝ j1 ,...,jm =1

whenever p > 2m/(m + 1). Let us prove the optimality of the exponents for p ≥ 2. Suppose that the theorem is valid for an exponent r, i.e.,

⎛

⎞ 1r ⏐r ⏐ ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ 1.3m0.365 ∥T ∥ . 1

n ∑

⎝ j1 ,...,jm =1

Since p ≥ 2, from the Kahane–Salem–Zygmund inequality ( Theorem 2.2) we have 1

m

n r ≤ 1.3m0.365 Cm n 2

) ( +m 12 − 1p

= Cm 1.3m0.365 n

mp+p−2m 2p

and thus, making n → ∞, we obtain r ≥ 2mp/(mp + p − 2m). For 1 < p ≤ 2, if the inequality holds for a certain exponent r, from the Kahane–Salem–Zygmund inequality (Theorem 2.2) we have 1− 1p

m

n r ≤ Cn

= Cn

p−1 p

and thus, making n → ∞, we obtain r ≥ mp/(p − 1). □ Remark 2.4. We stress that when 1 < p ≤ 2m/(m + 1), Theorem 2.3 does not provide a non-trivial r fulfilling the inequality (i.e., it may happen that the optimal r is infinity). Since

⎛

n ∑

⎝

p−2m ⎞ mp+2mp ⏐ 2mp ⏐ mp+p−2m ⏐T (ej , . . . , ejm )⏐ mp+p−2m ⎠ = (nm ) 2mp 1

j1 ,...,jm =1

for all unimodular m-linear forms T : ℓnp × · · · × ℓnp → K, as a consequence of the previous theorem we have the second main result of our paper, now addressed to the unbalancing lights problem: Corollary 2.5. If m ≥ 2 and n are positive integers and p ∈ [2, ∞], then mp+p−2m 1 n 2p , 1.3m0.365 and the exponents (mp + p − 2m)/2p are sharp.

K gm ,n (p) ≥

Remark 2.6. The puzzling nature of the Gale–Berlekamp switching game is a fertile ground for associated problems (see, for instance, [11,15]). Our result (that recovers the original problem when p = ∞) is another natural extension of this problem. It considers somewhat intermediate choices for the lights (not only on and off). We believe that the search for the optimal constants involved and the optimal exponents for the case p ∈ (1, 2) are natural and relevant topics for further investigation in this framework (see Fig. 1). The determination of the unknown exponents of Theorem 2.3 (and also Corollary 2.5) rely in an open result on the interpolation of certain multilinear forms, which seems to be open for a long time: it is well-known that every continuous m-linear form from ℓ1 ×· · ·×ℓ1 to K is multiple (1, 1)-summing and every continuous m-linear operators from ℓ2 ×· · ·×ℓ2 to K is multiple (2m/(m + 1), 1)-summing. What about intermediate results for ℓp ×· · ·×ℓp ? An interpolative approach (if it was possible) would

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

23

Fig. 1. Graphical overview of the exponents in Theorem 2.3.

tell us, for 1 < p < 2, that every continuous m-linear operators from ℓp × · · · × ℓp to K is multiple (mp/(m + p − 1), 1)-summing, and a simple computation as in the proof of Theorem 2.3 would tell us that the unknown exponents above would be in fact admissible exponents and give us the estimate K gm ,n (p) ≥ Km n

p−1 p

.

Even in the linear case, similar vector-valued problems remain open (see [9]). Based on the discussion of the previous paragraph we conjecture the following optimal result that also would complete the solution of Problem 1.3 with the remaining case 1 < p < 2: Conjecture 2.7. If m ≥ 2 and n are positive integers and p ∈ (1, 2), then there is a constant Km such that K gm ,n (p) ≥ Km n

p−1 p

and the exponents (p − 1)/p are sharp. We stress that as a consequence of Theorem 2.3 we already know that the exponent above cannot be smaller than (p − 1)/p. So just one direction of the conjecture remains to be proved. 3. Revisiting the classical unbalancing lights problem 3.1. The classical unbalancing lights problem For the sake of simplicity, in this section we shall now work with the case of real scalars as in the classical unbalancing lights problem, but the same arguments also work if we deal with complex scalars and coefficients with modulo 1. We shall prove a non asymptotic version of (1), showing all precise situations in which the sharp estimate is achieved. Theorem 3.1. Let aij = ±1 for 1 ≤ i, j ≤ n. Then there exist xi , yj = ±1, 1 ≤ i, j ≤ n, such that n ∑ i,j=1

1

3

aij xi yj ≥ 2− 2 n 2 ,

24

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

and the equality happens if, and only if, n = 2 and

[

( )

aij = ±

1 1

]

[

]

1 1 ,± −1 −1

[

1 1 ,± 1 1

−1 1

] or ±

[ −1 1

]

1 . 1

(9)

( )

In other words, for any initial configuration aij it is possible to perform switches so that the number ( ) of lights on minus the number of lights off is at least 2−1/2 n3/2 and the equality happens if and only if aij is as in (9). Proof. Littlewood’s 4/3-inequality asserts that

⎞ 34 n ∑ √ ⏐ ⏐4 ⏐T (ej , ek )⏐ 3 ⎠ ≤ 2 sup |T (x, y)| , ⎝ ⎛

(10)

∥x∥,∥y∥≤1

j,k=1

for all continuous bilinear forms T : ℓn∞ × ℓn∞ → R and all positive integers n. It is not difficult to prove that the supremum in the right-hand-side of (10) is achieved in the extreme points of the closed unit ball of ℓn∞ . Since these extreme point are precisely those with the entries 1 or −1, we conclude that there exist xi , yj = ±1, 1 ≤ i, j ≤ n, such that n ∑

1

3

aij xi yj ≥ 2− 2 n 2 .

i,j=1

( )

It remains to prove that the equality happens if and only if aij is as in (9). To prove this we recall the following result of [19]:

• A bilinear form T is an (norm-one) extreme of Littlewood’s 4/3 inequality if and only if T is written as 1

T (x, y) = ±2− 2 xi1 yi2 + xi1 yi3 + xi4 yi2 − xi4 yi3 , 1

T (x, y) = ±2− 2 1

T (x, y) = ±2− 2 1

T (x, y) = ±2− 2

) ( ) ( xi1 yi2 + xi1 yi3 − xi4 yi2 + xi4 yi3 , ) ( x y − xi1 yi3 + xi4 yi2 + xi4 yi3 , ) ( i1 i2 −xi1 yi2 + xi1 yi3 + xi4 yi2 + xi4 yi3

for i1 ̸ = i4 and i2 ̸ = i3 . From the above theorem we conclude that when we deal with bilinear forms with coefficients 1 or −1, the equality in (10) happens if and only if n = 2 and T (x, y) = ± (x1 y1 + x1 y2 + x2 y1 − x2 y2 ) , T (x, y) = ± (x1 y1 + x1 y2 − x2 y1 + x2 y2 ) , T (x, y) = ± (x1 y1 − x1 y2 + x2 y1 + x2 y2 ) , T (x, y) = ± (−x1 y1 + x1 y2 + x2 y1 + x2 y2 ) and the proof is done. □ 3.2. The classical unbalancing lights problem in higher dimensions The next result provides an asymptotic variant of (2) in the lines of (1): Theorem 3.2. Let m be a positive integer and ai1 ···im = ±1 for all i1 , . . . , im . Then, for all j = 1, . . . , m, (j) there exist xi = ±1, 1 ≤ i ≤ n, such that n ∑

⎛ 1

1

i1 ,...,im =1

1

(1) (m) ai1 ···im xi · · · xim ≥ ⎝2− 2 Ψ (m)− 2 γ

⎞ ( ( ) ) 2kk−2 m ∏ Γ 3k2k−2 m+1 (3) + o(1)⎠ n 2 , Γ 2 k=2

where Ψ is the digamma function and γ is the Euler–Mascheroni constant.

(11)

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

25

Before we prove Theorem 3.2, let us present some technical results. Lemma 3.3 (Minkowski). If 0 < p < q < ∞, then

⎛ ⎛ ⎞ 1q ×p ⎞ 1p ) 1p ×q ⎞ 1q n n ∑ ∑ ⏐ ⏐q ∑ ∑ ⏐ ⏐p ⏐aij ⏐ ⏐aij ⏐ ⎠ ⎟ ⎝ ⎠ ≤⎜ ⎝ ⎝ ⎠ ⎛

n

(

j=1

n

i=1

i=1

j=1

for all positive integers n and all scalars aij . Lemma 3.4 (Haagerup, see [18]). Let 1 ≤ p ≤ 2. For all sequence of real scalars (ai ) we have

(

n ∑

) 12 |ai |2

i=1

⎛ ⎞ ⏐p ) 1p ( p−2 ( p+1 ) )− 1p (∫ ⏐ n ⏐ 1 ⏐∑ ⏐ ⏐ ⎜ 2 2 Γ 2 ⎟ (3) ri (t)ai ⏐ dt ≤⎝ + o(1)⎠ , ⏐ ⏐ ⏐ Γ 2 0 k=1

where ri : [0, 1] → R are the Rademacher functions. Now we are able to proceed with the proof. Proof of Theorem 3.2. Let

( f (p) :=

2

( ) )− 1p Γ p+2 1 ( ) . Γ 23

p−2 2

Consider the m-linear form A x(1) , . . . , x(m) =

(

)

n ∑

(1)

(m)

ai1 ···im xi · · · xim . 1

i1 ,...,im =1

For bilinear forms, using Lemma 3.4, we have

)1 ( n )1 ( n n n ∑ ∑ ⏐ ⏐2 2 ∑ ∑⏐ ( )⏐2 2 ⏐aij ⏐ ⏐ A ei , ej ⏐ = j=1

i=1

j=1

i=1

⏐ n ⏐ ⏐∑ ( )⏐⏐ ⏐ ≤ (f (1) + o(1)) ri (t)A ei , ej ⏐ dt ⏐ ⏐ ⏐ j=1 0 i=1 ⏐ ( )⏐ n ⏐ n ⏐ ∑ ⏐ ∑ ⏐ ≤ (f (1) + o(1)) sup ri (t)ei , ej ⏐ ⏐A ⏐ ⏐ t ∈[0,1] n ∫ ∑

1

j=1

≤ (f (1) + o(1)) ∥A∥ .

i=1

(12)

and, by symmetry and by Lemma 3.3 we have

)2 ⎞ 12 ( n n ∑⏐ ⏐ ∑ ⏐aij ⏐ ⎠ ≤ (f (1) + o(1)) ∥A∥ . ⎝ ⎛

j=1

(13)

i=1

By the Hölder inequality for mixed sums (or interpolation as in [8]) combined with (12) and (13), we have

⎞ 34 n ∑ ⏐ ⏐4 ⏐aij ⏐ 3 ⎠ ≤ (f (1) + o(1)) ∥A∥ . ⎝ ⎛

i,j=1

26

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

For 3-linear forms we have

⎛ ⎛ ⎞ 12 × 43 ⎞ 43 n n ∑ ∑ ⏐ ⏐ ⎜ ⎟ ⏐aijk ⏐2 ⎠ ⎝ ⎝ ⎠ i,j=1

k=1

⎛ n ∫ ∑ ≤ (f (4/3) + o(1)) ⎝ k=1

1 0

⏐ n ⏐ 4 ⎞ 34 ⏐∑ ( )⏐⏐ 3 ⏐ rk (t)A ei , ej , ek ⏐ dt ⎠ ⏐ ⏐ ⏐ k=1

≤ (f (4/3) + o(1)) (f (1) + o(1)) ∥A∥ = (f (1)f (4/3) + o(1)) ∥A∥ .

(14)

Using symmetry and Lemma 3.3 we have

⎞ 34 ×2 ⎞ 12 ⎜ ∑ ⎝∑ ⏐⏐ ⏐⏐ 43 ⎠ ⎟ aijk ⎠ ≤ (f (1)f (4/3) + o(1)) ∥A∥ ⎝ ⎛

n

⎛

k,i=1

n

(15)

j=1

and

⎞1 ⎛ ⎞ 21 × 43 ⎞ 34 ×2 2 ⎛ n n n ⎜∑ ⎟ ⎟ ⎜∑ ⎝∑ ⏐⏐ ⏐⏐2 ⎠ ⎜ aijk ⎠ ⎟ ⎝ ⎠ ≤ (f (1)f (4/3) + o(1)) ∥A∥ . ⎝ ⎛

k=1

i=1

j=1

By the Hölder inequality for mixed sums and (14), (15), (16) we get

⎞ 32 n ∑ ⏐ ⏐3 ⏐aijk ⏐ 2 ⎠ ≤ (f (4/3) + o(1)) (f (1) + o(1)) ∥A∥ ⎝ ⎛

i,j,k=1

= (f (1)f (4/3) + o(1)) ∥A∥ . Following this vein, for the general case we have

⎛

+1 ⎞ m2m ) ) m ( ( ∏ ⏐ 2m 2 (k − 1) m+1 ⎠ ⏐ ∥A∥ ≤ f + o(1) 1 ···im

n ∑

⏐ ⏐ ai

⎝ i1 ,...,im =1

k

k=2

(( =

( )) m ∏ 2 (k − 1) f

k

k=2

) + o(1) ∥A∥ .

(j)

We thus conclude that there exist xi = ±1, 1 ≤ i ≤ n, such that n ∑

⎛(

(1)

(m)

ai1 ···im xi · · · xim 1

⎞ ( ))−1 m ∏ 2 (k − 1) m+1 ≥⎝ f + o(1)⎠ n 2

i1 ,...,im =1

k=2

k

⎞ ( 1 ( ) ) 2kk−2 m −2 ∏ 2− k Γ 3k2k m+1 (3) =⎝ + o(1)⎠ n 2 Γ 2 k=2 ⎛ ⎞ ( ( ) ) 2kk−2 m ∏ Γ 3k2k−2 1 m+1 1 Ψ (m) − γ − 2 (3) = ⎝2 2 + o(1)⎠ n 2 , Γ 2 k=2 ⎛

(16)

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

27

where Ψ is the digamma function and γ is the Euler–Mascheroni constant. The optimality of the exponent (m + 1)/2 can be proved, as usual, using the Kahane–Salem–Zygmund inequality. □ Observing that Lemma 3.4 holds for all sequence of real scalars (ai ), a close look to the proof of Theorem 3.2 tells us that in fact we have the following version, with asymptotic constants, of the Bohnenblust–Hille inequality: Theorem 3.5. For all continuous m-linear forms T : c0 × · · · × c0 → R we have

⎛

+1 ⎞ m2m ⏐ ( )⏐ m2m ⏐T ei , . . . , eim ⏐ +1 ⎠ 1

n ∑

⎝ i1 ,...,im =1

⎛ ≤ ⎝2

1 Ψ (m)+ 12 γ 2

m ∏

( Γ

k=2

⎞ ( ) ) 2kk−2 Γ 32 ( 3k−2 ) + o(1)⎠ ∥T ∥ ,

(17)

2k

where Ψ is the digamma function and γ is the Euler–Mascheroni constant. From (17) and repeating the proof of Theorem 2.3 we have: Theorem 3.6. Let p ∈ [2, ∞]. For all unimodular m-linear forms T : ℓnp × · · · × ℓnp → R we have

⎛

p−2m ⎞ mp+2mp ⏐ ( )⏐ mp+2mp ⏐T ei , . . . , eim ⏐ p−2m ⎠ 1

n ∑

⎝ i1 ,...,im =1

⎛ 1

1

≤ ⎝2 2 Ψ (m)+ 2 γ

m ∏ k=2

( Γ

⎞ ( ) ) 2kk−2 Γ 32 ( 3k−2 ) + o(1)⎠ ∥T ∥ ,

(18)

2k

where Ψ is the digamma function and γ is the Euler–Mascheroni constant.

1

1

Constant 2 2 Ψ (m)+ 2 γ m m m m m

=2 =5 = 10 = 100 = 1000

√ π/2 ≈ 1.2533 ≈ 1.7229 ≈ 2.2071 ≈ 5.0888 ≈ 11.7824

∏m

k=2

( Γ

Γ (

) 2kk−2

( ) 3 2

3k−2 2k

)

in (17) and (18)

4. Blow up rate for the Hardy–Littlewood inequalities for unimodular forms In this section we provide the blow up rate for the constants in Theorem 2.3 as n grows when the ℓ 2mp -norm in the left-hand-side is replaced by an ℓr -norm with 0 < r < ∞. This kind of mp+p−2m

approach has been previously addressed in [3,19] in a very similar situation. Also, the techniques that we shall address to prove the next result (Theorem 4.1) are the same of those used in [3,19]: essentially the Hölder inequality and the Kahane–Salem–Zygmund inequality combined with the estimates of Theorem 2.3. However, since now we are dealing just with unimodular multilinear forms, the optimal exponents of Theorem 2.3 are, in some cases, different from the optimal exponents of the Hardy– Littlewood inequalities (for instance, when p < 2m), and this is why some estimates here differ from those from [3]. More precisely, we prove the following result:

28

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

Theorem 4.1. If m is a positive integer and (r , p) ∈ (0, ∞) × (2m/(m + 1), ∞] then

⎛

⎞ 1r ⏐r ⏐ 2mr +2mp−mpr −pr ,0} ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ 1.3m0.365 nmax{ 2pr ∥T ∥ 1

n ∑

⎝ j1 ,...,jm =1

for all unimodular m-linear forms T : ℓnp × · · · × ℓnp → K and all positive integers n. Moreover, (1) For (r , p) ∈

((

0,

2mp mp+p−2m

)

) ([ ) ( ]) 2mp × [2, ∞] ∪ , ∞ × m2m , ∞ , the exponent max mp+p−2m +1

−mpr −pr { 2mr +2mp , 0} is sharp 2pr ( ) ( ) 2mp (2) For (r , p) ∈ 0, mp+p−2m × m2m , 2 the optimal exponent of n belongs to the interval +1 [ ] mp+r −pr 2mr +2mp−mpr −pr max{ pr , 0}, . 2pr

Proof. For p > 2m/(m + 1) we know from Theorem 2.3 that

⎛

p−2m ⎞ mp+2mp

n ∑

|T (ej1 , . . . , ejm )|

⎝

2mp mp+p−2m

≤ 1.3m0.365 ∥T ∥ .

⎠

(19)

j1 ,...,jm =1

(

Therefore, if (r , p) ∈ 0,

⎛

2mp mp+p−2m

)

×

(

2m m+1

] , ∞ , from Hölder’s inequality and (19) we have

⎞ 1r ⏐r ⏐ ⏐T (ej , . . . , ejm )⏐ ⎠ 1

n ∑

⎝ j1 ,...,jm =1

⎛ ≤⎝

p−2m ⎞ mp+2mp ⏐ 2mp ⏐ ⏐T (ej , . . . , ejm )⏐ mp+p−2m ⎠ 1

n ∑ j1 ,...,jm =1

⎛

n ∑

×⎝

−mpr −pr ⎞ 2mp+2mr 2mpr

|1|

2mpr 2mp+2mr −mpr −pr

⎠

j1 ,...,jm =1 −mpr −pr ( ) 2mp+2mr 2mpr ≤ 1.3m0.365 ∥T ∥ nm

= 1.3m0.365 n

2mr +2mp−mpr −pr 2pr

∥T ∥ .

(

Let us prove the optimality of the exponents for (r , p) ∈ 0,

2mp mp+p−2m

)

× [2, ∞]. Suppose that the

theorem is valid for an exponent s, i.e.,

⎛

n ∑

⎝

⎞ 1r ⏐ ⏐r ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ 1.3m0.365 ns ∥T ∥ . 1

j1 ,...,jm =1

Since p ≥ 2, from the Kahane–Salem–Zygmund inequality (Theorem 2.2) we have m

1

n r ≤ 1.3m0.365 ns Cm n 2

( ) +m 12 − 1p

= Cm 1.3m0.365 n

s+

mp+p−2m 2p

and thus, making n → ∞, we obtain s ≥ (2mr + 2mp − mpr − pr)/2pr.

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

29

[

) ( ] 2mp If (r , p) ∈ mp+p−2m , ∞ × m2m , ∞ we have (2mr + 2mp − mpr − pr)/2pr ≤ 0 and +1 ⎛

p−2m ⎞ mp+2mp ⏐ ⏐ 2mp ⏐T (ej , . . . , ejm )⏐ mp+p−2m ⎠ 1

⎛ ⎞ 1r n ∑ ⏐ ⏐r ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ ⎝ 1

n ∑

⎝

j1 ,...,jm =1

j1 ,...,jm =1

≤ 1.3m0.365 ∥T ∥ = 1.3m0.365 n

{

max

}

2mr +2mp−mpr −pr ,0 2pr

∥T ∥ . } 2mr +2mp−mpr −pr In this case the optimality of the exponent max , 0 is immediate, since no 2pr

{

negative exponent of n is ) possible. (

) , 2 , we just have an estimate for the optimal exponent of n. In fact, suppose that the inequalities are valid for an exponent s ≥ 0, i.e., ⎞ 1r ⎛ n ∑ ⏐r ⏐ ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ 1.3m0.365 ns ∥T ∥ . ⎝ 1 If (r , p) ∈ 0,

2mp mp+p−2m

×

(

2m m+1

j1 ,...,jm =1

Since 1 ≤

2m m+1

< p ≤ 2, from the Kahane–Salem–Zygmund inequality (Theorem 2.2) we have

m

n r ≤ 1.3m0.365 ns Cm n

1− 1p

= 1.3m0.365 Cm ns+

p−1 p

and thus, making n → ∞, we obtain s ≥ (mp + r − pr)/pr. □ If Conjecture 2.7 is correct, using the same ideas of the proof of the previous theorem it is possible to improve it to the following optimal result: Conjecture 4.2. Let m be a positive integer, (r , p) ∈ (0, ∞) × (1, ∞] and

⏐ mp + r − pr ⏐ , 0} ⏐t = max{ ⏐ pr ⏐ ⏐t = max{ 2mr + 2mp − mpr − pr , 0} ⏐ 2pr

for 1 < p ≤ 2, for p ≥ 2.

Then, there is a constant Km such that

⎛

n ∑

⎝

⎞ 1r ⏐ ⏐r ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ Km nt ∥T ∥ , 1

j1 ,...,jm =1

for all unimodular m-linear forms T : ℓnp × · · · × ℓnp → K and all positive integers n, and these values of t are sharp.

( In fact, ) the novelty is the case 1 < p ≤ 2. Supposing that Conjecture 2.7 is true, if (r , p) ∈ mp 0, p−1 × (1, 2], from Hölder’s inequality we have ⎛

n ∑

⎝

⎞ 1r ⏐ ⏐r mp+r −pr ⏐T (ej , . . . , ejm )⏐ ⎠ ≤ Km n pr ∥T ∥ . 1

j1 ,...,jm =1

On the other hand, if the above inequalities are valid for an exponent s instead of (mp + r − pr)/pr, since 1 < p ≤ 2, from the Kahane–Salem–Zygmund inequality (Theorem 2.2) we have m

n r ≤ Cns n

1− 1p

= Cns+

p−1 p

30

G. Araújo, D. Pellegrino / European Journal of Combinatorics 77 (2019) 17–30

and thus s ≥ (mp + r − pr)/pr . If (r , p) ∈ this case, the optimality of the exponent of n is possible.

[

) , ∞ × (1, 2] we have (mp + r − pr)/pr ≤ 0 and, in

mp p−1 mp+r −pr max pr

{

, 0} is immediate, since no negative exponent

Acknowledgments Part of this paper was written when D. Pellegrino was visiting Prof. F. Bayart at the Department of Mathematics of Université Blaise Pascal at Clermont Ferrand. He thanks Prof. Bayart for important comments and thanks the Réseau Franco-Brésilian en Mathématiques and CNPq, Brazil (Process 302834/2013-3) for the financial support. The authors are also very indebted to the two anonymous referees; their insightful and important suggestions were crucial to improve and clarify the presentation of the paper and to correct several imprecisions of the original version. References [1] N. Albuquerque, G. Araújo, M. Maia, T. Nogueira, D. Pellegrino, J. Santos, Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant, Ann. Math. Blaise Pascal 25 (1) (2018) 1–20, http://dx.doi.org/10.5802/ambp.371. [2] N. Alon, J. Spencer, The Probabilistic Method, Wiley, 1992, Second Edition, 2000, Third Edition 2008. [3] G. Araújo, D. Pellegrino, Optimal Hardy-Littlewood type inequalities for m-linear forms on ℓp spaces with 1 ≤ p ≤ m, Arch. Math. (Basel) 105 (3) (2015) 285–295. [4] G. Araújo, D. Pellegrino, On the constants of the Bohnenblust–Hille and Hardy–Littlewood inequalities, Bull. Braz. Math. Soc. (N.S.) 48 (1) (2017) 141–169. [5] G. Araújo, D. Pellegrino, D. Diniz P. Silva e Silva, On the upper bounds for the constants of the Hardy–Littlewood inequality, J. Funct. Anal. 267 (6) (2014) 1878–1888. [6] F. Bayart, Maximum modulus of random polynomials, Q. J. Math. 63 (1) (2012) 21–39. [7] F. Bayart, Multiple summing maps: Coordinatewise summability, inclusion theorems and p-Sidon sets, J. Funct. Anal. 274 (2018) 1129–1154. √ log n

[8] F. Bayart, D. Pellegrino, J. Seoane-Sepulveda, The Bohr radius of the n-dimensional polydisk is equivalent to , Adv. n Math. 264 (2014) 726–746. [9] G. Bennett, Schur multipliers, Duke Math. J. 44 (3) (1977) 603–639. [10] H.F. Bohnenblust, E, Hille On absolute convergence of Dirichlet series, Ann. Math. (2) 32 (1931) 600–622. [11] R.A. Brualdi, S.A. Meyer, Gale-Berlekamp permutation-switching problem, European J. Combin. 44 (2015) part A, 43–56. [12] A. Defant, P. Sevilla-Peris, The Bohnenblust-Hille cycle of ideas from a modern point of view, Funct. Approx. Comment. Math. 50 (1) (2014) 55–127. [13] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, 1995. [14] V. Dimant, P. Sevilla-Peris, Summation of coefficients of polynomials on ℓp spaces, Publ. Mat. 60 (2016) 289–310. [15] Y. Gordon, H.S. Witsenhausen, On extensions of the Gale-Berlekamp switching problem and constants of ℓp -spaces, Israel J. Math. 11 (1972) 216–229. [16] G. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Q. J. Math. 5 (1934) 241–254. [17] A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys. 53 (12) (2012) 122206, 15 pp. [18] F.L. Nazarov, A.N. Podkorytov, Ball, Haagerup, and Distribution Functions, in: Complex Analysis, Operators, and Related Topics. Operator Theory: Advances and Applications, Vol. 113, Havin, V.P. and Nikolski, N.K., Birkhäuser, Basel, 2000. [19] D. Pellegrino, E. Teixeira, Towards sharp Bohnenblust–Hille constants, Commun. Contemp. Math. 20 (3) (2018) 1750029, 33. [20] D. Popa, Multiple summing operators on ℓp spaces, Studia Math. 225 (1) (2014) 9–28. [21] T. Praciano-Pereira, On bounded multilinear forms on a class of ℓp spaces, J. Math. Anal. Appl. 81 (1981) 561–568.