Dense
المؤلف:
Bailey, D. H. and Crandall, R. E.
المصدر:
"On the Random Character of Fundamental Constant Expansions." Exper. Math. 10
الجزء والصفحة:
...
18-7-2021
2118
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 ![<span style=]()
{b^nalpha}" src="https://mathworld.wolfram.com/images/equations/Dense/Inline23.gif" style="height:15px; width:34px" /> 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.
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