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b-Dense Number

A real number alpha is b-dense if every possible finite string of consecutive digits occurs in the base-b expansion of alpha. If alpha is b-normal, then alpha is b-dense. If alpha is b-dense for some b, then alpha is an irrational number.

Equivalently, alpha is b-dense if the sequence of fractional parts {b^nalpha} is dense in the unit interval (Bailey and Crandall 2001, 2002).

See also

Dense, Dense Set, Normal Number

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References

Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001. https://doi.org/10.1080/10586458.2001.10504441.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002. https://doi.org/10.1080/10586458.2002.10504704.

Cite this as:

Weisstein, Eric W. "b-Dense Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/b-DenseNumber.html

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