A -automatic set is a set of integers whose
base-
representations form a regular language, i.e., a language accepted by a finite automaton
or state machine. If bases and are incompatible (do not have a common power) and if an -automatic set and -automatic set are both of density 0 over the integers, then it is believed
that
is finite. However, this problem has not been settled.

Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18,
... (OEIS A048645) have a simple arithmetic
expression. However, this is not the case for general -automatic sets.

Cobham, A. "On the Base-Dependence of Sets of Numbers Recognizable by Finite Automata." Math. Systems Th.3, 186-192,
1969.Cobham, A. "Uniform Tag Sequences." Math. Systems
Th.6, 164-192, 1972.Sloane, N. J. A. Sequence
A048645 in "The On-Line Encyclopedia
of Integer Sequences."