论文标题
欧几里得空间的大型子集避免了无限算术进行进程
Large subsets of Euclidean space avoiding infinite arithmetic progressions
论文作者
论文摘要
众所周知，如果$ \ mathbb {r} $的子集具有积极的lebesgue度量，则它包含任意长的有限算术进程。我们证明，从以下意义上讲，该结果并未扩展到无限的算术进程：对于$ [0,1）$中的每一个$λ$，我们构造了$ \ m athbb {r} $的子集，该子集与至少$λ$中的每个单位长度相交的每个间隔都与一组$λ$相交，但这并不包含任何无限的进程。
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.
