An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number $\beta$ such that $2^+\leq\beta\leq 8/3$, we show that there are arbitrarily long extremal $\beta$-free binary words.

中文翻译：

---

极值无重叠和极值 $\beta$-free 二进制字

如果通过在任何位置插入来自 $\Sigma$ 的单个字母从 $w$ 获得的每个单词包含一个在固定字母表 $\Sigma$ 上的无重叠（或 $\beta$-free）单词 $w$重叠（或至少为 $\beta$ 的指数因子）。我们找到所有允许极值无重叠二进制字的长度。对于每个扩展实数 $\beta$ 使得 $2^+\leq\beta\leq 8/3$，我们证明存在任意长的极值无 $\beta$ 二进制字。