On a sequence related to that of Thue-Morse and its applications

It is known that the sequence 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, . . . of lengths of blocks of identical symbols in the Thue–Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and 2. Its generating function W(x) is equal to ∏∞ k=1(1 + x(2 +(−1))/3). In terms of combinatorics on words, for any given x ∈ (0, 1) and > 0, we prove that every non-periodic word of an alphabet {1, 2} has a suffix s whose generating function S(x) satisfies the inequality xS(−x)> 1−W(−x)− . Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts { bn}, n= 0, 1, 2, . . ., where b<− 1 is a negative rational number and is a real number. Our results show, for example, that, for any real number = 0, the sequence of fractional parts { (−3/2)n}, n = 0, 1, 2, . . ., has a limit point greater than 0.466452. Furthermore, for each integer b −2 and each real number / ∈Q, we prove that lim infn→∞{ bn} ∏∞k=1(1−|b|−(2k+(−1)k−1)/3) and show that this inequality is sharp. © 2006 Elsevier B.V. All rights reserved. MSC: 11A63; 11K06; 68R15