A system of certain linear Diophantine equations on analogs of squares

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor \alpha n^2 \rfloor$ for $n\geq \alpha^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(\alpha)$, the set of all such tuples, is infinite for all $\alpha\in (0,1)\cap \mathbb{Q}$ and for almost all $\alpha\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $\alpha>0$ such that $T(\alpha)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $\lceil \alpha n^2 \rceil$ for $n\in \mathbb{N}$.