Large subsets of Euclidean space avoiding infinite arithmetic progressions
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 $λ$ in $[0,1)$, we construct a subset of $\mathbb{R}$ that intersects every interval of unit length in a set of measure at least $λ$, but that does not contain any infinite arithmetic progression.
math.CA↗