Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: For each α 2 there is an infinite binary word with critical exponent α. | For each a 2 there is an infinite binary word with critical exponent a James D. Currie Narad Rampersady Department of Mathematics and Statistics University of Winnipeg Winnipeg Manitoba R3B 2E9 CANADA e-mail Submitted Feb 28 2008 Accepted Aug 25 2008 Published Aug 31 2008 Mathematics Subject Classification 68R15 Abstract The critical exponent of an infinite word w is the supremum of all rational numbers a such that w contains an a-power. We resolve an open question of Krieger and Shallit by showing that for each a 2 there is an infinite binary word with critical exponent a. Keywords Combinatorics on words repetitions critical exponent 1 Introduction If a is a rational number a word w is an a-power if there exist words x and x and a positive integer n with x a prefix of x such that w xnx and a n x x . We refer to x as a period of w. A word is a-power-free if none of its subwords is a d-power with d a otherwise we say the word contains an a-power. The critical exponent of an infinite word w is defined as supfa 2 Q w contains an a-power . Critical exponents of certain classes of infinite words such as Sturmian words 8 10 and words generated by iterated morphisms 5 6 have received particular attention. Krieger and Shallit 7 proved that for every real number a 1 there is an infinite word with critical exponent a. As a tends to 1 the number of letters required to construct The author s research was supported by an NSERC operating grant. yThe author is supported by an NSERC Post-doctoral Fellowship. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N34 1 such words tends to infinity. However for a 7 3 Shur 9 gave a construction over a binary alphabet. For a 2 Krieger and Shallit gave a construction over a four-letter alphabet and left it as an open problem to determine if for every real number a 2 2 7 3 there is an infinite binary word with critical exponent a. Currie Rampersad and Shallit 3 gave examples of such words for a .
