On one additive problem connected with expansions on linear recurrrent sequence

Let $a_1,\ldots,a_d$ be natural numbers satisfying condition $a_1\geq a_2\geq\ldots\geq a_{d-1}\geq a_d=1.$ Define sequence $\{T_n\}$ using the linear recurrent relation $T_n=a_1T_{n-1}+a_2T_{n-2}+\ldots+a_dT_{n-d}$ and initial conditions $T_0=1,$ $T_n=1+a_1T_{n-1}+a_2T_{n-2}+\ldots+a_nT_0$ for $n<d$. Let $\mathbb{N}(w)$ be a set of natural numbers for which the greedy expansion on the linear recurrent sequence $\{T_n\}$ ends with some word $w$. Here $w$ is chosen in such a way that so that the set $\mathbb{N}(w)$ is non-empty. We study the problem about the number $r_k(N)$ of representations of a natural number $N$ in as the sum of $k$ terms from $\mathbb{N}(w)$.
Using the previously obtained description of the sets $\mathbb{N}(w)$ in terms of shifts of tori of dimension $d-1$, an asymptotic formula for the number of representations $r_k(N)$ is obtained, and also found upper bounds for the number of representations.
Conditions on $k$ that ensure the existence of considered representations for all sufficiently large natural numbers $N$ are found. In particular, such representations exist if $k\geq 1+(a_1+1)^{m-d+1}\frac{(a_1+1)^d-1}{a_1}$, where $m$ is the length of the fixed end $w$ of the greedy expansion. In addition, an asymptotic formula is obtained for the average number of representations.
In conclusion, several unsolved problems are formulated.