Wednesday, March 06, 2019

normal numbers and the books that can be written

Normal numbers are quite interesting - they have every substring of numbers in them. So an encoding of every book is in them, if you use, for example, ASCII. You'll get a sequence of numbers that encode every book somewhere in the number. They don't know if pi is a normal number, btw.

If you go far enough along the normal number's digits, you'll get a sequence that encodes any string in subsequent ASCII. For example, 90113  would encode "i am".

Where 9=i, 0=space, 1=a and 13=m

(There's a couple of "artificial" normal numbers that have been constructed. So they exist, and are computable.)

So a question occurred to me.

Say there's a place in the digits of a normal number where the encoding of a book starts. Call that B. For any book, B is going to be pretty big. Say, for example, the encoding of "the catcher in the rye" starts at the ten trillionth digit of this normal number - ASCII translation of the digits after 10 trillion encode that book, in other words.

Now let's define another number: the hopping number.

The hopping number is a step (say 7) you hop over ASCII encodings of letters to get to a new letter. Call that H. So you start at a new place, call that S, and you hop H, multiple times, each time noting the encoded letter to construct the entire book (again). Clearly, this sequence also encodes the book that started at B. So the question is, are there books where S is less than B?

Also, the ASCII encoding of the book must occur a number of times if the normal number is sufficiently large. How large?

It would have to be a function of S, N, B and H. Can we find that function?

Don't know. Do care.