On Gelfond-type problem for generalized Zeckendorf representations

Gelfond proved that for coprime $b-1$ and $d$ sums of digits of $b$-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference $d$. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences.
We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case $d=2$, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for $d=3$ the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers.
In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference $2$ with the logarithmic remainder term.