Given a real number 
,
the series
 |
is called the 
-expansion,
or 
-expansion (Parry 1957), of the positive
real number 
if, for all 
,
, where 
is the floor function
and 
need not be an integer. Any real number
such that 
has such an expansion, as can be found
using the greedy algorithm (Allouche and Cosnard
2000).
The special case of 
,
, and 
or 1 is sometimes called a 
-development (Komornik and Loreti 1998). 
gives the only 2-development. However, for almost all
, there are an infinite number
of different 
-developments.
Even more surprisingly though, there exist exceptional 
for which there exists only a single 
-development (Erdős et al. 1990, 1991, Komornik
and Loreti 1998). Furthermore, there is a smallest number 
known as the Komornik-Loreti
constant for which there exists a unique 
-development (Komornik and Loreti 1998).
." Acta. Math. Hungar. 58, 333-342,
1991.Erdős, P.; Joó, I.; and Komornik, V. "Characterization
of the Unique Expansions 
and Related Problems." Bull. Soc. Math. France 118, 377-390, 1990.Komornik,
V. and Loreti, P. "Unique Developments in Non-Integer Bases." Amer.
Math. Monthly 105, 636-639, 1998.Parry, W. "On the 
-Expansions of Real Numbers."
Acta Math. Acad. Sci. Hungar. 11, 401-416, 1960.Rényi,
A. "Representations for Real Numbers and Their Ergodic Properties." Acta
Math. Acad. Sci. Hungar. 8, 477-493, 1957.