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,…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 22. Its generating function W(x)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)x∈(0,1) and ε>0ε>0, we prove that every non-periodic word of an alphabet {1,2}{1,2} has a suffix ss whose generating function S(x)S(x) satisfies the inequality xS(-x)>1-W(-x)-εxS(-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}{ξbn}, n=0,1,2,…n=0,1,2,…, where b<-1b<-1 is a negative rational number and ξξ is a real number. Our results show, for example, that, for any real number ξ≠0ξ≠0, the sequence of fractional parts {ξ(-3/2)n}{ξ(-3/2)n}, n=0,1,2,…n=0,1,2,…, has a limit point greater than 0.4664520.466452. Furthermore, for each integer b⩽-2b⩽-2 and each real number ξ∉Qξ∉Q, we prove that liminfn→∞{ξbn}⩽∏k=1∞(1-|b|-(2k+(-1)k-1)/3) and show that this inequality is sharp.