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arXiv · 1310.4348

On the Union of Arithmetic Progressions

Shoni Gilboa·Rom Pinchasi
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Abstract

We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ 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 $c(\varepsilon)n^{2-\varepsilon}$ 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 $\ell$.

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Shoni Gilboa, Rom Pinchasi. 2013-10-16. On the Union of Arithmetic Progressions. https://doi.org/10.1137/130941122

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