A system of certain linear Diophantine equations on analogs of squares

Abstract

We investigate 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 the set $S(\alpha )$ of all integers of the form $\lfloor \alpha n^2 \rfloor $ for $n\geq \alpha ^{-1