An analogue of Eminian's problem for the Fibonacci number system

Gelfond proved the uniformity of distribution of the sums of binary digits expansions of natural numbers in arithmetic progressions. Later, this result was generalized to many other numeration systems, including Fibonacci numeration system.
Eminyan find an asymptotic formula for the number of natural $n$, not exceeding a given one, such that $n$ and $n+1$ have a given parity of the sum of digits of their binary expansions. Recently, this result was generalized by Shutov to the case of Fibonacci numeration system.
In the paper we consider quite more general problem about the number of natural $n$, not exceeding $X$, such that $n$ and $n+l$ have a given parity of the sum of digits of their representations in Fibonacci numeration system. A method is presented that allows to obtain asymptotic formula for a given quantity for all $l$. It is based on the study of some special sums associated with the problems and recurrence relations for these sums. It is shown that for any $l$ and all variants of parity the leading term of the asymptotic is different from the expected value $\frac{X}{4}$. Als it is proved that the remainder has the order $O(\log X)$. For $l\leq 10$ constants in the leading term of asymptotic formulas are found explicitly.
In the conclusion of the work, some open problems for further research are formulated.