Suppose $A\subseteq \mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\le C$ for all $n\in \mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\dots ,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\ast 1_{A_i}(n)<C$ for all $1\le i\le t$ and $n\in \mathbb{N}$?