Asymptotic contractivity of the Bohnenblust--Hille inequality for polynomials with few interacting variables

Let $K_{m,M}$ denote the optimal Bohnenblust--Hille constant on the class of $m$-homogeneous polynomials all of whose monomials involve at most $M$ different variables.

We prove that, for every fixed $M$, these constants are asymptotically contractive: \[ \lim_{m\to\infty}K_{m,M}=1. \] More precisely, \[ 1\le K_{m,M}\le A_M^{M/m}m^{(M^2-1)/(2m)}, \] where $A_M$ depends only on $M$.

The argument combines bounded projections onto exact support levels, a random colouring of the active variables, the classical multilinear Bohnenblust--Hille inequality and interpolation with Parseval's identity.

We also point out that the qualitative conclusion follows from a recent, more general support-sensitive Bohnenblust--Hille inequality, although its direct application gives a slightly larger power of the homogeneous degree.