Noncanonical number systems in the integers

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b| ≥ 4, and for b = −2. For the special case b = −2, we give a full characterisation of all valid digit sets.