On Thursday 03 November 2005 11:02, dasimov@earthlink.net wrote:
Some interesting ways to represent numbers in the interval (a,b):
Given (say positive) reals a < T < b, with T not of the form (K*a + L*b)/M for any integers K >= 0, L > = 0, M >= 1, create a sequence S_n of a's and b's as follows:
S_1 = a.
Let avg_n denote (S_1 + . . . + S_n) / n
If avg_n > T, then S_(n+1) = a; if avg_n < T, then S_(n+1) = b; if avg_n = T, the sequence terminates.
E.g., let a = 1, b = 2, and T = sqrt(2). Then what is the sequence of 1's and 2's ???
WLOG a=0,b=1. Then S_(n+1) is 1 iff the sum of the previous S_j is < nT. Let s_n be the sum of S_1..S_n; then we have s_1 = 0 s_(n+1) = s_n if s_n > nT s_(n+1) = s_n+1 if s_n < nT and by induction we can then show that s_n = ceiling((n-1)T). We get a 1 every time nT passes an integer.
E.g., let P_0 = 1. If (P_1*...*P_n) < e^n, then P_(n+1) = 3; else P_(n+1) = 2.
This last example brings up some questions: Is there a simple formula for P_n ? Are the lengths of runs on 2 or on 3 unbounded? Does the ratio
R_n = (# 2's in P_1 through P_n) / (# 3's in P_1 through P_n)
approach a limit as n -> oo ? If not, does log(R_n) remain bounded as n -> oo ? Does {R_n} have a lim sup or lim inf, and if so what are they?
In order (and in terms of the additive formulation with a=0,b=1): Is there a simple formula? Yes; see above. Are run lengths unbounded? No; if there is a run of k 0s then we must have no integer between nT and (n+k)T for some n, which implies that kT is small. Similarly for 1s. Does the ratio of the number of 0s and of 1s tend to a limit? Yes: (# 1s) / (# 0s + # 1s) = ceiling(nT)/n -> T, so (# 1s) / (# 0s) -> T/(1-T). -- g