Improving of estimate of the number of 6-aperiodic words of fixed length

W. Burnside raised the issue of locally finiteness of groups, all elements of which have finite order. A negative answer was received only in 1964 by E. S. Golod. Later S. V. Aleshin, R. I. Hryhorczuk, V. I. Sushchanskii proposed series of negative examples. Finiteness of the free Burnside group of period n installed at different times for n = 2, n = 3 (W. Burnside), n = 4 (W. Burnside, I. N. Sanov), n = 6 (M. Hall). Proof of infinity of this group for odd n ≥ 4381 was given by P. S. Novikov and S. I. Adian (1967), and for odd n ≥ 665 in the book of S. I. Adian (1975). More intuitive version of the proof for odd n > 1010 was proposed by A. Yu. Olshansky (1989). In connection with these results we consider sets of aperiodic words. Under the l-aperiodic word understand the word X if in it there is no non-empty subwords of the form Yl. We consider the question about the number of 2-aperiodic words in a two-letter alphabet and how many 3-aperiodic words in this alphabet. In the monograph of S. I. Adian (1975) shows a 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 of A. Yu. Olshansky (1989) a theorem on the infinity of the set of 6-aperiodic words and obtained a lower bound function for the number of words of a given length was proved. Our aim is to get more accurate estimate for the function of the number of 6-aperiodic words of given length. The results can be applied when encoding information is used in space communications.