Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung--Graham--Wilson theorem for graphs we provide a list of word properties which are equivalent to uniformity. In particular, we show that uniformity is equivalent to counting 3-letter subsequences. Inspired by graph limit theory we then investigate limits of convergent word sequences, those in which all subsequence densities converge. We show that convergent word sequences have a natural limit, namely Lebesgue measurable functions of the form $f:[0,1]\to[0,1]$. Via this theory we show that every hereditary word property is testable, address the problem of finite forcibility for word limits and establish as a byproduct a new model of random word sequences. Along the lines of the proof of the existence of word limits, we can also establish the existence of limits for higher dimensional structures. In particular, we obtain an alternative proof of the result by Hoppen, Kohayakawa, Moreira, R\'ath and Sampaio [{\it J. Combin. Theory Ser. B 103(1):93--113, 2013}] establishing the existence of permutons.

中文翻译：

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准随机词和词序列的限制

单词是有限字母表上的字母序列。我们为此对象研究两个密切相关的主题：准随机性和极限理论。关于第一个主题，我们研究了字母在区间上的均匀分布的概念，本着著名的 Chung--Graham--Wilson 定理的精神，我们提供了一个等价于均匀性的单词属性列表。特别是，我们证明了一致性相当于计算 3 个字母的子序列。受图极限理论的启发，我们研究了收敛词序列的极限，即所有子序列密度都收敛的词序列。我们证明了收敛词序列有一个自然极限，即 $f:[0,1]\to[0,1]$ 形式的 Lebesgue 可测函数。通过这个理论，我们证明了每个遗传词的属性都是可测试的，解决单词限制的有限强制问题，并作为副产品建立一个新的随机单词序列模型。沿着字数限制存在的证明，我们还可以建立更高维结构的限制存在。特别是，我们获得了 Hoppen、Kohayakawa、Moreira、R\'ath 和 Sampaio [{\it J. Combin. 理论系列 B 103(1):93--113, 2013}] 确定置换的存在。理论系列 B 103(1):93--113, 2013}] 确定置换的存在。理论系列 B 103(1):93--113, 2013}] 确定置换的存在。