The Champernowne constant, C10, can easily be formed by concatenating the natural numbers to produce the real number: 0.123456789101112... etc., see e.g. here.

But this is a bit wasteful, for as young Hannah Rollman (HR) has pointed out, we can afford to 'write down the numbers 1,2,3,... but omit any number (such as 12 or 23 or 31 ...) which has appeared as a string earlier in the sequence,' see e.g. here.

If we concatenate the HR numbers to produce a real number we obtain: 0.12345678910111314...etc. Call this the 'HR constant'.

It is known that C10 is "10-normal". (actually it was made up specifically so as to meet that criterion) i.e. in its decimal expansion the digit 7 must appear 1/10 of the time, the string 783 must appear 1/1000 of the time, and so on.

Question: is the HR constant "10-normal"?

(Inspired by thinking about 'Loops in Pi'.)

Rather than work in base 10, try base 2 for a moment, using the same method:

From 1 to 2: 110 (2*1,1*0)
From 3 to 4: 100 (2*0,1*1), 'neck and neck'
From 5 to 8: 1111000 (4*1,3*0)
From 9 to 16: 101110000 (5*0,4*1), 'neck and neck' again.
From 17 to 32: 100101010110110110011110111111100000 (15*0,21*1)
From 33 to 64: 1000111001001001101010001011001110101110111000000 (27*0,23*1)
with 1 barely ahead, but since the string 17 to 32 contains 1111111, this discrepancy is likely only temporary.

Using base 3, the string is: 12101120221001021101111222002012121000 (15*1,12*0,11*2); but again the early preponderance of 1's is liable to be reversed  by the elision of numbers such as eg. 1111,1112.

Query whether there might even be something like a constructive proof along these lines.

Edited on July 2, 2012, 7:14 am

Posted by broll on 2012-07-02 06:16:29