b-Dense Number

A -dense number is a real number whose expansion in an integer base contains every finite string of consecutive digits, including strings with leading zeros. Such a number is also called disjunctive in base (Leśniak 2014, p. 150). If is -normal, then is -dense. If is -dense for some , then is an irrational number.

Equivalently, is -dense iff the sequence of fractional parts ${b^nalpha}$ , for , is dense in , where ${y}=y-|_y_|$ and is the floor function (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.Leśniak, K. "On Discrete Stochastic Processes with Disjunctive Outcomes." Bull. Austral. Math. Soc. 90, 149-159, 2014. https://doi.org/10.1017/S0004972714000124.

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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