The Zeckendorf representation holds for integers-and pi isn't an integer; so the assumption seems incorrect and, thus, the conclusions don't follow.  However, already for integers, why would the fact that the summands converge to a Gaussian imply something about the distribution of the digits of the number itself? This is a separate problem. 

You should read book "Fibonacci and Lucas numbers with application" by Koshy paragraph "Fiboanacci and Pi" page 95. You fing the exact formula for pi as limit of logaritm of Fobonacci numbers.

The papers to which you point only state results "on the average": not for given numbers.  As such, they might suggest things that are true about pi, but are unlikely to prove anything (especially given how hard it is to show that any particular real number is normal to lots of bases, even though it's almost trivial to show that almost all numbers are normal to all bases).

Actually, no-the quoted papers prove results about the distribution of the summands; the point is that the relation of these results, for integers, to results about, apparently, their logarithms and, hence, to pi, seems harder to grasp. 

Quoting from the abstract of the first link: "average number of summands needed for integers in [F_n,F_{n+1})"  It's not a limit for a particular sequence of values of n, it's for a particular sequence of averages over intervals depending on n.  At least, that is my reading of the abstract of the first link.

The Zeckendorf representation of an integer is its unique representation as a sum of non-consecutive Fibonacci numbers. So we can not apply it for pi, as it is not an integer (not even a rational number).

It is believed that Pi is a normal number, but I think,  a convincing proof is yet to come..

There are ways, however, to write real numbers as sums of non-consecutive powers of phi: it's no longer unique (any more than decimal expansions of reals are unique): but it is still interesting.

The question of course is about the digits of Pi.  To a given precision the sequence is an integer e.g. 141592653589793238462643383279502884197169399375105820974944592307816406286... etc. with a unique Zekendorf representation. 

For n digits you have a Zn presentation with the coefficients {z1..zN} while for n+1 digits you have Zn+1 etc. If you were to describe the probability for the nth digit given its representation for the previous digits it would follow the normal distribution etc. Which means that Pi's digits in the Zekendorf code are randomly distributed.   

Asher: there is nothing in the papers cited that says anything about a particular sequence of integers. Indeed, your statement above could be made for any real number: for example, 0.  And it would clearly be wrong in that case: it may well be correct when made about the sequence of convergents to pi that you are taking, but it is likely to be immensely hard to prove.  The digits of pi are not random: they are a fixed sequence.  The statements in the papers are all of the form "if you take a random integer selected uniformly in the range [F_n, F_[n+1}], this is how its Zeckendorf representation behaves."  As I said above, the digits of pi are not a random integer, and neither are they constrained to fall into the appropriate intervals.  

Now, if you have techniques that can *prove* something about the nth digit of pi given its earlier digits, you will have a very very nice result.  But I think that it would surprise  the pi community greatly if you do (as was the case, for example, with the Bailey Borwein Plouffe algorithm --- which only exists in bases which are powers of 2, not in base 10). 

@Neil

Check this out: http://www.huffingtonpost.com/david-h-bailey/are-the-digits-of-pi-random_b_3085725.html?

Yes --- I know Jon and David's paper well.  It is certainly the case that pi seems to *look* normal in every base.  But that is far from a proof.

While the digits of π are, obviously, generated by a well-defined procedure, the statement about their distribution doesn't imply that they are random in that sense. It is, of course, possible to define, within any given subsequence of the digits of pi, a frequency measure and, simply, count, within a subsequence of size N, how many times, N_d, each decimal digit appears and then try to study how the ratio N_d/N behaves as N increases. Or one may ask the question of correlations between digits: if a 3 appears in position k, of a subsequence of size N, how does that affect the appearance of a 4, for instance in position l? And how do these results depend on the position of the subsequence in the decimal expansion. And so on. 

Therefore, indeed, if the statement is, one studies the fractional part of π in this way, then the Zeckendorf representation may be useful in this way. 

@Neil, @Stam

I guess that the proof is connected to question (4) in section 1.4 of  Miller & Wang's paper. 

"What can we say about the distribution of the largest gap between summands in

the Zeckendorf decomposition? Appropriately normalized, how does the distribution

of gaps between the summands behave? What is the distribution of the

largest gap? How often is there a gap of 2?"

If the gaps are distributed randomly I would argue that the Zekendorf code is random. 

The statement ``the gaps are distributed randomly'' isn't precise enough-and doesn't express the quotation. What Miller and Wang state there is what I recalled in a previous message-that they can, apparently, compute the distribution, not just some of its moments. And this distribution need not be uniform-nor Gaussian. However what isn't clear is in what way *this* distribution is related to the distribution of the digits of π and thus can answer the question, whether the distribution of π's digits is normal. 

Let's look at the Z'dorf code for the sequence 14159 -- 10010010000000101001 and compare to the next sequence 141592 -- 1000100001010100010100001, add the next digit and you get 100000010100100101000001010101. You can keep going to a higher number of digits... Here is a sequence of 10 digits, 1415926535 -- 10010000001010000100001001010100010101000010. I find the gaps and in particular the largest gap to be very demonstrative. 

Maybe-but, of course, the question is to quantify this, to check that it isn't an artifact and develop the theory.

One way to quantify up to N digits precision is to histogram the Z'dorf gaps for π vs. √2, Φ, e, etc.  and compare the distributions. One can use the BBP formula (David Bailey, Peter Borwein, and Simon Plouffe) to generate the Nth digit for π. For the rest of the beasts I assume that Robert Nemiroff and Jerry Bonnell could help. What will follow this effort is some statistical analysis which may lead to exciting results. 

Attached is a file depicting the histograms above as computed for 1000 digits of the above numbers. I played around with fitting the normalized Z'dorf gaps for π to the Benford's law. Plots are added. I assume that going to a much larger number of digits will tell more but the indications so far are interesting. 

Also attached is the associated Mathematica code. Please let me know if you find a problem. 

BPP is really only useful for binary computations.  There are much better ways to get decimal digits, and one can easily compute way more than 1000 digits in realistic time.  

I have not been trying to suggest that looking at the Zeckendorf representations is uninteresting --- far from it --- but rather that it seems very unlikely to me that you'll be able to use the results in the papers that you cite to prove anything about the randomness of pi.

According to http://arxiv.org/pdf/1402.3912v2.pdf the gap distribution (gap k to k+1) should follow a geometric decay. I find for 5000 digits a perfect exponential decay ~ 0.6 e-0.47 k   -- whose discrete analog is the geometric decay.  More statistics can establish the decay constant f(t) = λ e−λt. Assuming that the gaps are distributed geometrically does this in any way help in the randomness argument? Can one argue that for  n → ∞ digits the Zeckendorf code as characterized by its gap distribution is a random variable hence the digits must be random?         

http://arxiv.org/pdf/1402.3912v2.pdf

Not yet. What must, first, be established is that the exponential decay found for the 5000 digits, doesn't depend on the position of the sequence of 5000 digits; next, how does it behave as one takes longer and longer sequences, try to find a pattern and prove the pattern. These aren't easy steps.

Right. It is consistent with the 1000 digits etc. The pattern is in the statistics which need my attention now... 

Thanks to Steven J. Miller who opened my eyes to the fact that you cannot conclude anything about decompositions coming from a special sequence. The geometric decay is similar to every sequence. This closes this little thread.