On the number of points of an incomplete lattice in rectangular regions

In 2013–2015 it was shown that for any purely real algebraic irrationality $\alpha$, starting from some place, all residual fractions in the expansion of $\alpha$ into a continued fraction will appear to be the reduced algebraic irrationalities.
We construct the examples of purely real algebraic irrationalities $\alpha$ for which this number of the residual fraction is arbitrarily large.