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q-Expansion


Given a real number q>1, the series

 x=sum_(n=0)^inftya_nq^(-n)

is called the q-expansion, or beta-expansion (Parry 1957), of the positive real number x if, for all n>=0, 0<=a_n<=|_q_|, where |_q_| is the floor function and a_n need not be an integer. Any real number x such that 0<=x<=q|_q_|/(q-1) has such an expansion, as can be found using the greedy algorithm (Allouche and Cosnard 2000).

The special case of x=1, a_0=0, and a_n=0 or 1 is sometimes called a q-development (Komornik and Loreti 1998). a_n=1 gives the only 2-development. However, for almost all 1<q<2, there are an infinite number of different q-developments. Even more surprisingly though, there exist exceptional q in (0,1) for which there exists only a single q-development (Erdős et al. 1990, 1991, Komornik and Loreti 1998). Furthermore, there is a smallest number 1<q<2 known as the Komornik-Loreti constant for which there exists a unique q-development (Komornik and Loreti 1998).


See also

Komornik-Loreti Constant, Pisot Number, Salem Constants

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References

Allouche, J.-P. and Cosnard, M. "The Komornik-Loreti Constant Is Transcendental." Amer. Math. Monthly 107, 448-449, 2000.Erdős, P.; Horváth, M.; and Joó, I. "On the Uniqueness of the Expansions 1=sumq^(-n_i)." Acta. Math. Hungar. 58, 333-342, 1991.Erdős, P.; Joó, I.; and Komornik, V. "Characterization of the Unique Expansions q=sumq^(-n_i) 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 beta-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.

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q-Expansion

Cite this as:

Weisstein, Eric W. "q-Expansion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-Expansion.html

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