An automatic set in base 
, where 
is an integer, is a set
of nonnegative integers whose base-
representations form a regular
language, i.e., a formal language accepted
by a finite automaton. Such a set is also called
-automatic.
Two bases 
are multiplicatively
independent if 
for all positive integers 
and 
. An automatic set is called sparse if its number of elements
at most 
is 
as 
for some nonnegative
integer 
,
where 
denotes big-O notation. Albayrak and Bell (2023)
proved that a sparse 
-automatic
set and a sparse 
-automatic
set have finite intersection
when the bases are multiplicatively
independent. They conjectured that the intersection
is still finite when one set is
sparse and the other has natural density zero.
For example, the 2-automatic set of positive integers whose binary representations contain one or two 1s is
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645).
This is the union of 
and 
, where 
and 
are integers.
