6-aperiodic words over the three-letter alphabet

The work is devoted to the study of sets of aperiodic words over a finite alphabet. A set of such words can be considered as some kind of finite formal language. W. Burnside raised the issue of local finiteness of periodic groups. The negative answer was given only sixty years later by E. S. Golod. Soon S. V. Aleshin, R. I. Hryhorczuk, V. I. Sushchanskii constructed more examples confirming the negative answer to Burnside's question. Finiteness of the free Burnside group of period n was established for periods two and three (W. Burnside), for period four (W. Burnside, I. N. Sanov), for period six (M. Hall). The infinity of such a group, for odd indicators exceeding 4381, is established in the work of P. S. Novikov and S. I. Adyan (1967), and for odd indicators exceeding 664 in the book by S. I. Adian (1975). A more intuitive version of the proof for odd n > 10¹⁰ was proposed by A. Yu. Olshansky (1989). In this article, we consider the set of 6-aperiodic words. In the monograph by S. I. Adyan (1975) it was shown the proof of S. E. Arshon (1937) theory that there are infinitely many three-aperiodic words of any length in the two-letter alphabet. In the book of A. Y. Olshansky (1989), a proof of the infinity of the set of six-aperiodic words is given and an estimate of the number of such words of any given length is obtained. Here we try to estimate the function of the number of six-aperiodic words of any given length in a three-letter alphabet. The results obtained can be useful for encoding information in space communication sessions.