Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Newman proved for the classical Thue–Morse sequence, ( ( − 1 ) s ( n ) ) n ≥ 0 , that c 1 N λ < ∑ n = 0 N − 1 ( − 1 ) s ( 3 n ) < c 2 N λ for all N ∈ N with real constants λ , c 1 , c 2 satisfying c 2 > c 1 > 0 and λ = log 3 / log 4 . Coquet improved this result and deduced ∑ n = 0 N − 1 ( − 1 ) s ( 3 n ) = N λ F ( log 4 N ) + η ( N ) 3 , where F ( x ) is a nowhere-differentiable, continuous function with period 1 and η ( N ) ∈ { − 1 , 0 , 1 } . In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum ∑ n = 0 N − 1 ( − 1 ) s γ ( 3 n + r ) a Coquet-type formula exists for every r ∈ { 0 , 1 , 2 } if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.