论文标题
避免无限算术 /几何进程的大型集
Large sets avoiding infinite arithmetic / geometric progressions
论文作者
论文摘要
我们研究了ERD相似性问题的一些变体。我们提出一个问题，如果实际线的每个可测量子集都包含一个无限几何进程的类似副本。我们构建了真实行的紧凑型子集$ e $，因此$ 0 $是$ e $的lebesgue密度点，但是$ e $不包含任何（非恒定）无限的几何发展。我们提供了足够的密度类型条件，可以保证一组包含无限的几何进程。 通过稍微改善Bradford，Kohut和Mooroogen arXiv的最新结果：2205.04786，我们构建了一个封闭的套装$ f \ subset [0，\ infty）$，因此，$ f \ cap cap [t，t+1] $的量度倾向于$ 1 $，但$ f $ in Infinity却不包含任何不知名的进程。对于更多的一般序列，我们还略微改善了Kolountzakis和Papageorgiou Arxiv的最新结果：2208.02637。 我们提供了足够的条件，可以保证给定的cantor类型集包含至少一个无限的几何发展，任何商在$ 0 $和$ 1 $之间。这可以应用于大多数对称的正态度量集。
We study some variants of the Erdős similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen arXiv:2205.04786, we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou arXiv:2208.02637 for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.
