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.
