We investigate the problem of the maximum number of cubic subwords (of theform $www$) in a given word. We also consider square subwords (of the form$ww$). The problem of the maximum number of squares in a word is not wellunderstood. Several new results related to this problem are produced in thepaper. We consider two simple problems related to the maximum number ofsubwords which are squares or which are highly repetitive; then we provide anontrivial estimation for the number of cubes. We show that the maximum numberof squares $xx$ such that $x$ is not a primitive word (nonprimitive squares) ina word of length $n$ is exactly $\lfloor \frac{n}{2}\rfloor - 1$, and themaximum number of subwords of the form $x^k$, for $k\ge 3$, is exactly $n-2$.In particular, the maximum number of cubes in a word is not greater than $n-2$either. Using very technical properties of occurrences of cubes, we improvethis bound significantly. We show that the maximum number of cubes in a word oflength $n$ is between $(1/2)n$ and $(4/5)n$. (In particular, we improve thelower bound from the conference version of the paper.)
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机译：我们研究给定单词中立方子单词（以$ www $形式）的最大数目的问题。我们还考虑了方形子字（形式为$ ww $）。一个单词中最大平方数的问题还没有得到很好的理解。本文产生了与该问题有关的几个新结果。我们考虑两个与子词的最大数目有关的简单问题，子词为正方形或高度重复。然后我们对立方体的数量提供非平凡的估计。我们表明，长度为$ n $的单词中，使得$ x $不是原始单词（非原始正方形）的最大平方数$ xx $恰好是$ \ lfloor \ frac {n} {2} \ rfloor-1 $，对于$ k \ ge 3 $，形式为$ x ^ k $的子词的最大数量恰好是$ n-2 $。尤其是一个单词中的多维数据集的最大数量不大于$ n-2 $要么。利用多维数据集出现的非常技术性的属性，我们可以显着改善此边界。我们表明，长度为$ n $的单词中的多维数据集的最大数量在$（1/2）n $和$（4/5）n $之间。 （特别是，我们改进了会议版本的下限。）
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