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

Newman proved for the classical Thue–Morse sequence, ((−1)s(n))n≥0((−1)s(n))n≥0, that c1Nλ<∑n=0N−1(−1)s(3n)c1>0c2>c1>0 and λ=log3/log4λ=log3/log4. Coquet improved this result and deduced ∑n=0N−1(−1)s(3n)=NλF(log4N)+η(N)3, where F(x)F(x) is a nowhere-differentiable, continuous function with period 11 and η(N)∈{−1,0,1}η(N)∈{−1,0,1}. In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum ∑n=0N−1(−1)sγ(3n+r) a Coquet-type formula   exists for every r∈{0,1,2}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.