# Does every sufficiently long string contain many repetitions of a string of bounded length?

Let $$S$$ be a finite set and $$d > 0$$. Does there exist $$\ell > 0$$ such that the following holds?

Every sufficiently long string with letters in $$S$$ contains at least $$d$$ consecutive copies of some string of length at most $$\ell$$.

For example, when $$S = \{0, 1\}$$ and $$d = 2$$, we can take $$\ell = 2$$: every string of length at least $$4$$ contains one of $$00, 11, 0101, 1010$$.

No, since the Prouhet–Thue–Morse sequence $$t$$ is an infinite cube-free sequence on a two-letter alphabet.
Thus, for each $$N > 0$$, the prefix of length $$N$$ of $$t$$ contains no factor of the form $$uuu$$, with $$|u| > 0$$, and a fortiori no factor of the form $$u^d$$, with $$d \geqslant 3$$, of any length.