Newman's phenomenon for generalized Thue-Morse sequences
by
Thomas Stoll
University of Waterloo

Let t_j=(-1)^s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman says that t₀+t₃+t₆+...+ t_3k > 0 for all k ≥ 0. We show that t₁+t₄ +t₇+...+t_3k+1 < 0 and t₂+t₅ +t₈+...+ t_3k+2 ≤ 0 for k ≥ 0, where equality is characterized by means of an automaton. This sharpens results given by Dumont. More general settings are studied. For a, g ≥ 2 let w_a=exp(2pi/a) and t^{(a, g)}_j=w_a^s_g(j), where s_g(j) denotes the sum of digits in the g-ary digit expansion of j. We show that the case a=2 inherits many Newman-like phenomena for every even g ≥ 2 and large classes of arithmetic progressions of indices. Similar results also hold in the case a=3. This, in particular, extends results by Drmota and Skaaba to the general g-case.

Date received: June 10, 2008