On the sum of digits of expansions of a pair of consecutive numbers over a linear recurrent sequence (Q6084898)

Let \(a_1, \ldots, a_d\) be positive integers satisfying the condition \[ a_1 \geq a_2 \geq \cdots \geq a_{d-1} \geq a_d=1. \] Define a sequence \(\left\{T_n\right\}\) using a linear recurrent relation \[ T_n=a_1 T_{n-1}+a_2 T_{n-2}+\cdots+a_d T_{n-d}. \] The initial conditions have the form \[ T_0=1, \quad T_n=1+a_1 T_{n-1}+a_2 T_{n-2}+\cdots+a_n T_0 \] for \(n<d\). In this case, any positive integer \(N\) admits a unique greedy expansion with respect to the sequence \(\left\{T_n\right\}\): \[ N=\sum_{k=0}^{m(N)} \varepsilon_k(N) T_k. \] This expansion being greedy means that the inequalities \(0 \leq N-\sum_{k=m_1}^{m(N)} \varepsilon_k(N) T_k<T_{m_1}\) hold for any \(m_1<m(N)\). Define the sets \[ \mathscr{N}_0=\left\{n: \sum_{k=0}^{m(N)} \varepsilon_k(N) \equiv 0\pmod 2\right\}, \quad \mathscr{N}_1=\left\{n: \sum_{k=0}^{m(N)} \varepsilon_k(N) \equiv 1\pmod 2\right\} \] of positive integers with a given parity of the sum of the digits of the expansion with respect to the sequence \(\left\{T_n\right\}\). Let \[ T_{i, j}(X)=\sharp\left\{n \leq X: n \in \mathscr{N}_i, n+1 \in \mathscr{N}_j\right\}. \] The main result of the paper is the following theorem. Theorem. There exist effectively computable \(\lambda, 0<\lambda<1\), and \(C_{i j}\left(C_{00}=C_{11}=-C_{10}=-C_{01}\right)\) such that \[ T_{i, j}(X)=\left(\frac{1}{4}+C_{i j}\right) X+O\left(X^\lambda\right). \] For the special case in which \(\left\{T_n\right\}\) is a Fibonacci sequence \(\left(d=2, a_1=a_2=1\right)\), this problem was considered by the author in [Fibonacci Q. 58, No. 3, 203--207 (2020; Zbl 1468.11025); Dal'nevost. Mat. Zh. 20, No. 2, 271--275 (2020; Zbl 1484.11061)] where it was shown in two different ways that, in this case, \[ \begin{aligned} & T_{i, j}(X)=\frac{\sqrt{5}}{10} X+O(\log X) \quad \text { for } \quad i=j, \\ & T_{i, j}(X)=\frac{5-\sqrt{5}}{10} X+O(\log X) \quad \text { for } \quad i \neq j . \end{aligned} \]