Fici et al. defined a word to be a k-power if it is the concatenation of k consecutive identical blocks, and an r-antipower if it is the concatenation of r pairwise distinct blocks of the same size. They defined $N(k,r)$ as the smallest ℓ such that every binary word of length ℓ contains either a k-power or an r-antipower. In this note we obtain some new upper and lower bounds on $N(k,r)$. We also consider avoiding 3-antipowers and 4-antipowers over larger alphabets, and obtain a lower bound for $N(k,5)$ in the binary case.

中文翻译：

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反权力的新界限

Fici等。定义一个字是一个ķ -power如果它的串联ķ连续相同的块，和- [R -antipower如果是的串联- [R相同尺寸的成对不同块。他们定义$ñ（ķ，\mathrm{[R}）$作为最小的ℓ，使得每个长度为ℓ的二进制字都包含k次幂或r次幂。在本说明中，我们获得了一些新的上下界$ñ（ķ，\mathrm{[R}）$。我们还考虑在较大的字母上避免使用3反幂和4反幂，并为$ñ（ķ，5）$ 在二进制情况下。