Dense

A set  in a first-countable space is dense in  if , where  is the set of limit points of . For example, the rational numbers are dense in the reals. In general, a subset  of  is dense if its set closure .

A real number  is said to be -dense iff, in the base- expansion of , every possible finite string of consecutive digits appears. If  is -normal, then  is also -dense. If, for some ,  is -dense, then  is irrational. Finally,  is -dense iff the sequence  is dense (Bailey and Crandall 2001, 2003).

REFERENCES:

Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001.

Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.