An ergodic approach to equations of the form $x+y=\lfloorα(n)\rfloor$

Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloorα(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $α(t)$ is a sufficiently regul