A repetition in a word is a pair of words and such that is a factor of , is non-empty, and is a prefix of . If is a repetition, then its period is and its exponent is . A word is -free (resp. -free) if it contains no repetition with exponent such that (resp. ).
Given , Dejean  defined the repetition threshold for letters as the smallest such that there exists an infinite -free word over a -letter alphabet. Dejean initiated the study of in 1972 for and . Her work was followed by a series of papers which determine the exact value of for any .
Mousavi and Shallit  have considered two notions related to the repetition threshold.
The first notion considers repetitions in conjugates of factors of the infinite word. A word is circularly -free if it does not contain a factor such that is a repetition of exponent strictly greater than . Let . The smallest real number such that is circularly -free is denoted by . Let be the minimum of over every .
The second notion considers repetitions in concatenations of a fixed number of factors of the infinite word. Let be the smallest real number such that every product of factors of is -free. Let be the minimum of over every . Notice that generalizes the classical notion of repetition threshold which corresponds to the case , that is, for every .
Our first result shows that the case corresponds to the first notion of repetition avoidance in conjugates.
for every .
Mousavi and Shallit  have considered the binary alphabet and obtained that for every . Our second result considers the ternary alphabet and gives the value of for every . This extends the result of Dejean  that and the result of Mousavi and Shallit  that .
if or is even.
if is odd and .
Proof of Theorem 1.
The language of words in avoiding circular repetitions of exponent at least (or strictly greater than ) is a factorial language. As it is well-known , if a factorial language is infinite, then it contains a uniformly recurrent word . By Proposition 14 in , . This implies that .
To obtain the two equalities of Theorem 2, we show the two lower bounds and then the two upper bounds.
Proof of for every even .
Mousavi and Shallit  have proved that , which settles the case . We have double checked their computation of the lower bound . Suppose that is a fixed even integer and that is an infinite ternary word. The lower bound for implies that there exists two factors and such that with . Thus, the prefix of is also a product of two factors of . So we can form the -terms product which is a repetition of the form with exponent . This is the desired lower bound.
Proof of for every odd .
Suppose that is a fixed odd integer, that is, . Suppose that is a recurrent ternary word such that the product of factors of is never a repetition of exponent at least . First, is square-free since otherwise there would exist an -terms product of exponent . Also, does not contain two factors and with the following properties:
Indeed, this would produce the -terms product which is a repetition of the form with exponent .
So if , , and are distinct letters, then does not contain both and and does not contain both and . A computer check shows that no infinite ternary square-free word satisfies this property. This proves the desired lower bound.
Proof of for every even .
Let be any even integer at least . To prove this upper bound, it is sufficient to construct a ternary word satisfying . The ternary morphic word used in  to obtain seems to satisfy the property. However, it is easier for us to consider another construction. Let us show that the image of every -free word over by the following -uniform morphism satisfies .
Recall that a word is -free if it does not contain a repetition with period at least and exponent strictly greater than . First, we check that such ternary images are -free using the method in . By Lemma 2.1 in , it is sufficient to check this freeness property for the image of every -free word over of length smaller than . Since , the period of every repetition formed from pieces and with exponent at least must be at most . Then we check exhaustively by computer that the ternary images do not contain two factors and such that
Thus, the period of every repetition formed from pieces and with exponent strictly greater than must be at most . So we only need to check that for -terms products that are repetitions of period at most .
Now the period is bounded, but can still be arbitrarily large, a priori. For every factor of length at most , we define as the length of a largest factor of that is a -terms product, divided by . We actually consider conjugacy classes, since if is a conjugate of , then . Let be such a factor. If, for some even , we have , then it means that by appending a -terms product to a -terms product that corresponds to a maximum factor of , that can only add a cube of period . This implies that for every , .
We have checked by computer that for every conjugacy class of words of length at most , there exists a (small) even such that . Thus we have in all cases.
Proof of for every odd .
Let us show that the image of every -free word over by the following -uniform morphism satisfies for every odd .
First, we check that such ternary images are -free using the method in . By Lemma 2.1 in , it is sufficient to check this freeness property for the image of every -free word over of length smaller than . Thus, the period of every repetition formed from pieces and with exponent strictly greater than must be at most . Using the same argument as in the previous proof, we have checked by computer that for every conjugacy class of words of length at most , there exists a (small) odd such that . Thus we have in all cases.
3 Concluding remarks
The next step would be to consider the -letter alphabet. Obviously, for every and . Mousavi and Shallit  verified that , so that for every . We conjecture that this is best possible, i.e., that for every . However, a proof of an upper bound of the form cannot be similar to the proof of the upper bounds of Theorem 2. The multiplicative factor of , which drops from when to when , forbids that the constructed word is the morphic image of any (unspecified) Dejean word over a given alphabet.
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