Theoretically? Citation needed. I believe this is only true if pi is normal, and while there's a lot of conjecture that pi is normal and no evidence that suggests it isn't, there is no proof.
Normal or not normal? If a work can be found in pi, then one would think a Finite State Gambler (FSG) would succeed on the portion of pi equivalent to the work. But FSGs cannot succeed on normal sequences, meaning this sequence would be non-normal.
Of course this begs the question of whether a number normal over an infinite sequence can contain sub-sequences that are non-normal - sequences of 100 tails will occur in sufficiently large completely normal (randomized) sequences of random coin tosses, e.g.
If this is correct, then you are correct, the conjecture would hold if pi were normal. But then wouldn't an FSG be able to succeed on the sub-sequence? Of is that acceptable, provided that the FSG fails on the overall sequence?
Normality is sufficient, but I do not think it is necessary. Normality requires that if the number is represented in any base b then each each of the b possible digits occure with density 1/b. All that is needed for every work to appear is that each digit occur at least once. They do not have to occure with equal density.
Normality is not necessary. Counter-counter example is the digits of pi in groups of 1, 2, 3... with strings of zeros in between:
3.0140015900026530000....
this number is obviously nor normal since 0 occurs with greater frequency than any other digit. But if pi is normal, then this sequence contains all finite sequences too.
Each base 10 digit occurs at least once in your number. It's not obvious to me that each base b digit occurs at least once for all bases b in your number.
Comments
Theoretically? Citation needed. I believe this is only true if pi is normal, and while there's a lot of conjecture that pi is normal and no evidence that suggests it isn't, there is no proof.
Normal or not normal? If a work can be found in pi, then one would think a Finite State Gambler (FSG) would succeed on the portion of pi equivalent to the work. But FSGs cannot succeed on normal sequences, meaning this sequence would be non-normal.
Of course this begs the question of whether a number normal over an infinite sequence can contain sub-sequences that are non-normal - sequences of 100 tails will occur in sufficiently large completely normal (randomized) sequences of random coin tosses, e.g.
A normal sequence includes all non-normal sequences, by definition of normality.
As for the citation, there is this: http://en.wikipedia.org/wiki/Disjunctive_sequence
If this is correct, then you are correct, the conjecture would hold if pi were normal. But then wouldn't an FSG be able to succeed on the sub-sequence? Of is that acceptable, provided that the FSG fails on the overall sequence?
Normality is sufficient, but I do not think it is necessary. Normality requires that if the number is represented in any base b then each each of the b possible digits occure with density 1/b. All that is needed for every work to appear is that each digit occur at least once. They do not have to occure with equal density.
Nope, normality is necessary.
Counter example would be a=0,123456789010011000111000011110000011111....
Normality is not necessary. Counter-counter example is the digits of pi in groups of 1, 2, 3... with strings of zeros in between:
this number is obviously nor normal since 0 occurs with greater frequency than any other digit. But if pi is normal, then this sequence contains all finite sequences too.Each base 10 digit occurs at least once in your number. It's not obvious to me that each base b digit occurs at least once for all bases b in your number.