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

AbstractIt 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(2k+(-1)k-1)/3). In terms of combinatorics on words, for any given x∈(0,1) and ε>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)-ε. Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts {ξbn}, n=0,1,2,…, where b<-1 is a negative rational number and ξ is a real number. Our results show, for example, that, for any real number ξ≠0, the sequence of fractional parts {ξ(-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 ξ∉Q, we prove that liminfn→∞{ξbn}⩽∏k=1∞(1-|b|-(2k+(-1)k-1)/3) and show that this inequality is sharp