Good point, Rich. I was thinking of the partial sums Sum{k=1...n} exp(n i c) as an analogy, but this is a bad analogy. --Dan << Since almost all C are normal numbers, I'd imagine that the partial sums resemble a random walk. And that the walk gradually wanders away from the origin, at distance O(sqrt(step#)). Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov [dasimov@earthlink.net] Sent: Wednesday, December 10, 2008 11:06 AM To: math-fun Subject: Re: [math-fun] Twubbling Turtle Trajectories (HAKMEM 45) Marc asks what if anything new is now known re this Hakmem question. Special cases such as those discussed below will probably present the most difficulty. But I would also guess that for almost all c in [0,1], the set of values P(c) = {2^n c mod 1 | n = 1,2,3,...} is uniformly distributed in [0,1] -- and that (hence?) the set of partial sums { Sum{k=1...n} exp(2^k c 2pi i) | n = 1,2,3,...} is a bounded set for almost all c (i.e., with probability 1). ---------------------------------------------------- (On the other hand, the last I heard is that the set {(3/2)^n mod 1 | n = 1,2,3,...} is not even known to be dense (!) in [0,1].)

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_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele