Dan Asimov <dasimov@earthlink.net> wrote:

Do you mean normal to a specific base?

No. To all integer bases B > 1.

I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases.

Scott Huddleston <c.scott.huddleston@gmail.com> wrote:

How do you define normal to factorial base?

I'm assuming that by "factorial base" Dan means this: http://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values Since the Nth place has N allowed values, defining normality is tricky. I'd say if you divide those N values into B equal bins in each case, to create a base-B number, if that base-B number is normal for all integers B > 1, I'd say that counts as normal. For instance suppose the Nth digit of your factorial base number is the Nth prime mod N (Sloane's A004648): 0, 1, 2, 3, 1, 1, 3, 3, 5, 9, 9, 1, 2, 1, 2, ... Divide each of them by N: 0/1, 1/2, 2/3, 3/4, 1/5, 1/6, 3/7, 3/8, 5/9, 9/10, 9/11, 1/12, 2/13, 1/15, 2/15, ... Bin these fractions into, say, deciles (B=10), i.e. the first decimal digit of the fraction: 0, 5, 6, 7, 2, 1, 4, 3, 5, 9, 8, 0, 1, 0, 1, ... If 0.056721435980101... is normal in base 10, and similarly with all other values of B, I'd say the original number is normal to the factorial base. This also works for defining what's meant by the normality of an infinite list of real numbers that are between 0 and 1. However, I think a more natural form of normality than base factorial is the continued fraction form of a number. It's normal if the coefficients have the expected distribution, in which each coefficient appears less often than the next smallest coefficient, and in which the geometric mean of the coefficients approaches Khinchin's constant. Granted, this isn't quite the same thing. For example quadratic irrationals are all probably normal in every integer base, but *none* of them are normal in the continued fraction sense.