THERE EXIST MULTILINEAR BOHNENBLUST–HILLE CONSTANTS (C n ) ∞ n=1 WITH lim n→∞ (C n+1 − C n )=0 D. NU ˜ NEZ-ALARC ´ ON * , D. PELLEGRINO ** , J.B. SEOANE-SEP ´ ULVEDA *** , AND D. M. SERRANO-RODR ´ IGUEZ * Abstract. The n-linear Bohnenblust–Hille inequality asserts that there is a con- stant C n ∈ [1, ∞) such that the ℓ 2n n+1 -norm of ( U (e i1 ,...,e in ) ) N i1,...,in=1 is bounded above by C n times the supremum norm of U, regardless of the n-linear form U : C N ×···× C N → C and the positive integer N (the same holds for real scalars). The power 2n/(n + 1) is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponential growth. Very recently, a new panorama emerged and the impor- tance, for many applications, of the knowledge of the optimal constants (denoted by (K n ) ∞ n=1 ) was stressed. The title of this paper is part of our Fundamental Lemma, one of the novelties presented here. It brings new (and precise) information on the optimal constants (for both real and complex scalars). For instance, K n+1 − K n < 0.87 n 0.473 for infinitely many n’s. In the case of complex scalars we present a curious formula, where π,e and the famous Euler–Mascheroni constant γ appear together: K n < 1+   4 √ π  1 − e γ/2-1/2  n-1  j=1 j log 2( e -γ/2+1/2 ) -1   for all n ≥ 2. Numerically, the above formula shows a surprising low growth, K n < 1.41 (n − 1) 0.305 − 0.04 for every integer n ≥ 2. We also provide a brief discussion on the interplay be- tween the Kahane–Salem–Zygmund and the Bohnenblust–Hille (polynomial and multilinear) inequalities. We shall adapt some of the techniques presented here to estimate the constants satisfying Bohnenblust–Hille type inequalities when the exponent 2n n+1 is replaced by any q ∈  2n n+1 , ∞  . 2010 Mathematics Subject Classification. 46G25, 30B50. Key words and phrases. Bohnenblust–Hille inequality, Kahane–Salem–Zygmund inequality, Quantum Information Theory. * Supported by Capes. ** Supported by CNPq Grant 301237/2009-3. *** Supported by grant MTM2009-07848. 1 arXiv:1207.0124v5 [math.FA] 14 Sep 2012