Covering shrinking polynomials by quasi progressions

Abstract

Erd\H os introduced the quantity $S=T{\sum }_{i=1}^T{X}_i$, 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. Sárközy confirmed this conjecture and proved that $S\ge cN/{\log }^{2}N$. In this paper, we extend this to shrinking polynomials and so-called $\{X_i\}$ quasi progressions.

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