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arXiv · 2205.04786

Large subsets of Euclidean space avoiding infinite arithmetic progressions

Laurestine Bradford·Hannah Kohut·Yuveshen Mooroogen
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Abstract

It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each $\lambda$ in $[0,1)$, we construct a subset of $\mathbb{R}$ that intersects every interval of unit length in a set of measure at least $\lambda$, but that does not contain any infinite arithmetic progression.

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Laurestine Bradford, Hannah Kohut, Yuveshen Mooroogen. 2022-05-10. Large subsets of Euclidean space avoiding infinite arithmetic progressions. https://arxiv.org/abs/2205.04786

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