康托尔集中的等差数列间隙集
Arithmetic-progression gap sets in Cantor sets
AI总结：
该研究探讨分形集（尤其是满足强分离的仿射自相似集、中间$\varepsilon$康托尔集）中等差数列的公差限制，证明了临界参数下最大等差级数长度的突变，并得出了公差的相关区间结果。
AI中文摘要：
我们研究分形集中的等差数列可出现哪些公差的问题。对于紧集$C\subset\mathbb R$，我们不仅考察$C$中是否存在等差数列，还考察其所有公差构成的集合。更一般地，对于有限模式$P$，我们研究仿射拷贝$P$在$C$中出现的尺度集合。对于满足强分离的仿射自相似集，我们得到了可容许公差的明确限制。针对中间$\varepsilon$康托尔集，我们证明当$3-2\sqrt2<\varepsilon\le 1/3$时，最长等差数列的长度为4，这表明在临界参数$\varepsilon=3-2\sqrt2$处，最大等差级数长度会立即从6下降。我们进一步推导了可容许公差集合的递归界，并得出了明确的“消隐区间”，即等差数列无法出现的尺度范围。从正面结果来看，足够厚的康托尔集表现出相反的行为。结合Hunt-Kan-Yorke构造的改进版本与Newhouse间隙引理，我们证明每个足够小的公差都出现在三项等差数列中。特别地，若康托尔集的最大有界间隙不超过$0.067 diam(C)$，且其厚度至少为$6.96268\ldots$，则区间$(0,0.435 diam(C)]$中的每个公差都出现在$C$包含的三项等差数列中。针对四项等差数列和非对称三点模式也得到了类似的区间结果。
英文摘要：
We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set $C\subset\mathbb R$, we investigate not only whether arithmetic progressions occur in $C$, but the full collection of their common differences. More generally, for a finite pattern $P$, we study the set of scales at which affine copies of $P$ appear in $C$. For affine self-similar sets satisfying strong separation, we obtain explicit restrictions on admissible common differences. Specializing to middle-$\varepsilon$ Cantor sets, we prove that the longest arithmetic progression has length four whenever $3-2\sqrt2<\varepsilon\le 1/3$, showing that the maximal progression length drops immediately from six at the critical parameter $\varepsilon=3-2\sqrt2$. We further develop recursive bounds for the sets of admissible common differences and derive explicit blackout intervals, namely ranges of scales for which arithmetic progressions cannot occur. On the positive side, sufficiently thick Cantor sets exhibit the opposite behavior. Combining a refinement of the Hunt-Kan-Yorke construction with the Newhouse Gap Lemma, we prove that every sufficiently small common difference occurs in a three-term arithmetic progression. In particular, if the largest bounded gap of a Cantor set is at most $0.067 diam(C)$ and its thickness is at least $6.96268\ldots$, then every common difference in $(0,0.435 diam(C)]$ occurs in a three-term arithmetic progression contained in $C$. Analogous interval results are obtained for four-term arithmetic progressions and asymmetric three-point patterns.
