# Normal vs disjunctive vs lexicon

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?)

Sub-questions:

• 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.