Estimating the number of 12-aperiodic words

In 1902 W. Burnside raised the issue of local finiteness of groups, all elements of which are of finite order. A negative answer was obtained only 63 years later by E. S. Golod. Then S. V. Aleshin, R. I. Hryhorczuk, V.I. Sushchanskii proposed a series of negative examples. Finiteness of the free Burnside group of period n was established for n = 2, n = 3 (W. Burnside), n = 4 (W. Burnside, I. N. Sanov), n = 6 (M. Hall). The proof of infinity of this group for odd n ≥ 4381 was given in the article by P. S. Novikov and S. I. Adian (1967), and for odd n ≥ 665 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). For n = 12 the answer is still unknown. A. S. Mamontov installed local finiteness of the group of period 12 without the elements of order 12. This result generalizes Theorems of I. N. Sanov and M. Hall. D. V. Lytkina, V. D. Mazurov and A. S. Mamontov proved that the group of period 12, in which the order of the product of any two elements of order two is not greater than 4, is locally finite. This theorem generalizes Theorem of I. N. Sanov, where the group of period 12 without elements of order 6 is locally finite. In relation with these results we consider the set of 12-aperiodic words. The word is called l-aperiodic if there are no non-empty subwords of the form Yl in it. In the monograph by S. I. Adian (1975) it was shown the proof of S. E. Arshon (1937) of the fact that in the two letters alphabet there is an infinite set of arbitrarily long 3-aperiodic words. In the book by A. Yu. Olshansky (1989) the theorem on the infinity of the set of 6-aperiodic words was proved, and a lower bound function for the number of words of a given length was obtained. Our aim is to get an estimate for the function of the number of 12-aperiodic words of the length n. The results can be applied when encoding information in space communications.