This question is directly inspired by "Does Pi contain all possible number combinations?". I would like to state firstly for the record that I have no serious number theory education. I think I probably could solve all exercises from Elements of Number Theory by Vinogradov given enough time but that is pretty much my limit.
Non the less I have given some thought to similar questions. They pup up in my subject (Dynamical Systems) on several places. I learned about weakly and strongly irrational numbers from trying to understand proofs (yes Arnold and Moser proofs are different and they do not prove the identical statements) of KAM theorem. All possible combinations argument appears when one uses symbolic dynamics to show existence of transitive orbit in let say logistical map.
$\pi$ is just an example of irrational number. As every student of Calculus in U.S. learned during the fist semester each real number has a decimal representation. If the number is irrational that decimal representations contains infinite number of digits without repeating patterns. So the natural questions are:
- Does a decimal representation of a weakly irrational number contain
all possible number combinations? - Does a decimal representation of a
strongly irrational number contain all possible number combinations? - Does a decimal representation of an algebraic irrational number
contain all possible number combinations? - Does a decimal representation of a transcendental irrational number contain all possible number combinations ($\pi$ is just a special case)?
Best Answer
The current state is that determing if an irratioal number contains every finite sequence is completely out of reach.
Every algebraic irrational number and numbers like $\pi$ and $e$ are widely believed to be even NORMAL ( which is a stronger property ), but there is no irrational number which was not specially constructed for which it could be proven or disproven, that it contains every finite sequence.