Density of a-powers in repetitions of a primitive word

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We consider words over fixed alphabet. A word 푢 is said to be primitive if it cannot be expressed as a nontrivial power of another word. For a positive real 훼 > 1 the 훼-power of a word 푣 is defined by 푣 훼 = 푣 b훼c 푝 where 푝 is a prefix of 푣 of length (훼− b훼c)|푣| . We prove that for any primitive word 푢 and for any 훼 > 1 the number of distinct 훼-powers in 푢 푘 is asymptotically equal to 1 훼 |푢 푘 | for 푘 → ∞.