  Erd\H os introduced the quantity $S=T\sum^T_{i=1}X_i$, where $X_1,\dots, X_T$
are arithmetic progressions, and cover the square numbers up to $N$. He
conjectured that $S$ is close to $N$, i.e. the square numbers cannot be covered
"economically" by arithmetic progressions. S\'ark\"ozy 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.
