On the growth of the optimal constants of the multilinear Bohnenblust--Hille inequality

Let $(K_n{)}_{n=1}^{\mathrm{\infty }}$ be the optimal constants satisfying the multilinear (real or complex) Bohnenblust–Hille inequality. The exact values of the constants $K_n$ are still waiting to be discovered since eighty years ago; recently, it was proved that $(K_n{)}_{n=1}^{\mathrm{\infty }}$ has a subexponential growth. In this note we go a step further and address the following question: Is it true that $$\underset{n\to \mathrm{\infty }}{\lim }(K_n-K_{n-1})=0?$$ Our main result is a Dichotomy Theorem for the constants satisfying the Bohnenblust–Hille inequality; in particular we show that the answer to the above problem is essentially positive in a sense that will be clear along the note. Another consequence of the dichotomy proved in this note is that $(K_n{)}_{n=1}^{\mathrm{\infty }}$ has a kind of subpolynomial growth: if $p(n)$ is any non-constant polynomial, then $K_n$ $\mathit{\text{is}}$ $\mathit{\text{not}}$ asymptotically equal to $p(n)\mathrm{.}$ Moreover, if $$q>{\log }_2(\frac{e^{1-1\mathrm{/}2\gamma }}{\sqrt{2}})\approx 0.526,$$ then $$K_n\nsim n^q\mathrm{.}$$