Sets whose differences avoid a bracket quadratic

Suppose a set of integers $A\subseteq \{1,\dots ,N\}$ has no solutions to $a-a^{\mathrm{\prime }}=n\lfloor \sqrt[3]{2}n\rfloor ,$ for distinct $a,a^{\mathrm{\prime }}\in A,$ and $n\in \mathbb{N}\mathrm{.}$ We show that $\mathrm{\mid }A\mathrm{\mid }\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).