Apologies for lack of rigour but I'll attempt to phrase this in an answerable way. In this question, @Charles writes:

[Being a normal number] (or even the weaker property of being disjunctive) implies that every possible string occurs somewhere in its expansion.

As I understand a normal number:

an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base-b is often called b-normal.

And obviously normality can be base-dependent.

Whereas a disjunctive number is:

a real number whose expansion with respect to some base b is a disjunctive sequence over the alphabet {0,...,b−1}

And a lexicon:

So, in a given base, it's true to sat that being disjunctive is weaker than being normal (and this is also base-dependent)?

A number that is disjunctive to every base is called absolutely disjunctive

Main question:

If a number contains "every possible string somewhere in its expansion" it is normal (and therefore disjunctive). Is having an infinite expansion and a statistically random expansive enough to say a number is disjunctive (and therefore contain every possible string somewhere in its expansion?)


  • For each of these properties (normal, disjunctive, lexicon), is it true to say that the expansion in at least one base is infinite? I assume it must be.
  • What about all bases?
  • It seems that random numbers (defined by using strong randomness tests) are normal. For each of these properties (normal, disjunctive, lexicon), is it true to say the rough reverse? That the expansion is statistically random over {0, ..., b-1} in base b? (explicitly ignoring predictability as a measure of randomness such as pi or Champernowne's number) c.f. Borel, E., Rend. Circ. Mat. Palermo 27 (1909) p. 247.)
  • Simply normal numbers are a subset of disjunctive numbers. Is the same true for normal numbers?
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