On sum-product bases

Abstract

Besides various asymptotic results on the concept of sum-product bases in ${\mathbb{N}}_0$, we consider by probabilistic arguments the existence of thin sets $A,A^{\prime }$ of integers such that $AA+A={\mathbb{N}}_0$ and $A^{\prime }{A}^{\prime }+A^{\prime }{A}^{\prime }={\mathbb{N}}_0$.