On the Union of Arithmetic Progressions

Abstract

We show that for every $\epsilon >0$ there is an absolute constant $c(\epsilon )>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at least $c(\epsilon )n^{2-\epsilon }$ elements. We observe, by construction, that one can find $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences such that the cardinality of their union is $o(n^2)$. We refer also to the non-symmetric case of $n$ arithmetic progressions, each of length $\ell$, for various regimes of $n$ and $\ell$.