  This paper studies left invertibility of discrete-time linear I/O quantized
linear systems of dimension 1. Quantized outputs are generated according to a
given partition of the state-space, while inputs are sequences on a finite
alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left
D-invertibility, under suitable conditions. While left invertibility takes into
account membership in sets of a given partition, left D-invertibility considers
only distances, and is very easy to detect. Considering the system $x^+=ax+u$,
our main result states that left invertibility and left D-invertibility are
equivalent, for all but a (computable) set of $a$'s, discrete except for the
possible presence of two accumulation point. In other words, from a practical
point of view left invertibility and left D--invertibility are equivalent
except for a finite number of cases. The proof of this equivalence involves
some number theoretic techniques that have revealed a mathematical problem
important in itself. Finally, some examples are presented to show the
application of the proposed method.
