Power-Saving Bounds For Monic Minkowski Polynomials

We prove that if $f\in \mathbb{Z}[x]$ is a monic polynomial of degree $k\ge 2$, then there exists a constant $c>0$, depending only on $f$, and finite sets $A\subset \mathbb{R}$ of arbitrarily large size such that $$\mathrm{\mid }f(A)\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{k-c},$$ where $f(A)$ is interpreted in the Minkowski sum-product sense. In particular, taking $f(x)=x^2+x$, this gives a power-saving upper bound for $AA+A$, answering a question raised by Roche-Newton, Ruzsa, Shen, and Shkredov.