Covering shrinking polynomials by quasi progressions

Erdős introduced the quantity $S=T\sum^T_{i=1}|X_i|$, where $X_1,\dots, X_T$ are arithmetic progressions that cover the squares up to $N$. He conjectured that $S$ is close to $N$, i.e. the square numbers cannot be covered "economically" by arithmetic progressions. Sárközy confirmed this conjecture and proved that $S\geq cN/\log^2N$. In this paper we extend this to shrinking polynomials and so-called $\{X_i\}$ quasi progressions.
Passcode: a six digit number $N=(4!)^2+(p-5)^2$ where $p$ is the smallest prime such that $p>600.$