Newman's phenomenon for generalized Thue–Morse sequences

Let tj=(-1)s(j)tj=(-1)s(j) be the Thue–Morse sequence with s(j)s(j) denoting the sum of the digits in the binary expansion of j  . A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719–721] says that t0+t3+t6+⋯+t3k>0t0+t3+t6+⋯+t3k>0 for all k⩾0k⩾0.In the first part of the paper we show that t1+t4+t7+⋯+t3k+1<0t1+t4+t7+⋯+t3k+1<0 and t2+t5+t8+⋯+t3k+2⩽0t2+t5+t8+⋯+t3k+2⩽0 for k⩾0k⩾0, where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145–148]. In the second part we study more general settings. For a,g⩾2a,g⩾2 let ωa=exp(2πi/a)ωa=exp(2πi/a) and tj(a,g)=ωasg(j), where sg(j)sg(j) denotes the sum of digits in the g-ary digit expansion of j  . We observe trivial Newman-like phenomena whenever a|(g-1)a|(g-1). Furthermore, we show that the case a=2a=2 inherits many Newman-like phenomena for every even g⩾2g⩾2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Skałba [Rarified sums of the Thue–Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609–642] to the general g-case.