Sets whose differences avoid a bracket quadratic

Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certa