Let the root of the word $w$ be the smallest prefix $v$ of $w$ such that $w$ is a prefix of $vvv...$. $per(w)$ is the length of the root of $w$. For any $n\ge5$, an $n$-ary threshold word is a word $w$ such that for any factor (subword) $v$ of $w$ the condition $\frac{|v|}{per(v)}\le\frac{n}{n-1}$ holds. Dejean conjecture (completely proven in 2009) states for $n\ge5$ that exists infinitely many of $n$-ary TWs. This manuscript is based on the author's student works (diplomas of 2011 (bachelor's thesis) and 2013 (master's thesis) years) and presents an edited version (in Russian) of these works with some improvements. In a 2011 work proposed new methods of proving of the Dejean conjecture for some odd cases $n\ge5$, using computer verification in polynomial time (depending on $n$). Moreover, the constructed threshold words (TWs) are ciclic/ring TWs (any cyclic shift is a TW). In the 2013 work, the proof method (of 2011) was improved by reducing the verification conditions. A solution for some even cases $n\ge6$ is also proposed. A 2013 work also proposed a method to construct stronger TWs, using a TW tree with regular exponential growth. Namely, the TWs, where all long factors have an exponent close to 1.

翻译：令词 $w$ 的根为其最小前缀 $v$，使得 $w$ 是 $vvv...$ 的前缀。$per(w)$ 表示 $w$ 的根的长度。对于任意 $n\ge5$，一个 $n$ 元阈值词是指满足以下条件的词 $w$：对于 $w$ 的任意因子（子词）$v$，条件 $\frac{|v|}{per(v)}\le\frac{n}{n-1}$ 均成立。德让猜想（于2009年被完全证明）指出对于 $n\ge5$，存在无限多个 $n$ 元阈值词。本手稿基于作者的学生工作（2011年（学士论文）和2013年（硕士论文）的学位论文），并呈现了这些工作的修订版（俄语），其中包含若干改进。在2011年的工作中，提出了证明德让猜想在某些奇数情况 $n\ge5$ 下成立的新方法，该方法使用多项式时间（依赖于 $n$）的计算机验证。此外，所构造的阈值词是循环/环形阈值词（任意循环移位仍为阈值词）。在2013年的工作中，通过减少验证条件改进了（2011年的）证明方法。同时提出了针对某些偶数情况 $n\ge6$ 的解决方案。2013年的工作还提出了一种构造更强阈值词的方法，该方法利用具有正则指数增长的阈值词树。具体而言，这些阈值词中所有长因子的指数均接近1。