Covering shrinking polynomials by quasi progressions

TL;DR

This paper extends Erdős's conjecture on covering square numbers with arithmetic progressions to include shrinking polynomials and quasi progressions, providing new bounds and insights into their covering properties.

Contribution

It generalizes the covering problem from arithmetic progressions to shrinking polynomials and quasi progressions, establishing new bounds and extending previous conjectures.

Findings

Extended Erdős's covering bounds to shrinking polynomials

Proved that quasi progressions have similar covering limitations

Established lower bounds for covering square numbers with these sequences

Abstract

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.