Re: [math-fun] Random integers
Michael Collins wrote: ----- I think the "symmetry" argument is destroyed by the non-constructive choice of A; i.e. I suspect there are some A which would assign non-zero probability to 37, some which select an even integer with probability other than 1/2 et cetera. ----- No. Regardless of what problems this method may have, asymmetry between different integers is not one of them. Recap of method: ---------------- Identify the circle R/Z with the half-open interval [0, 1). Then R/Z is partitioned into subsets {A_n | n in Z} whose disjoint union is R/Z and which are all rotated versions of each other. That allows this theoretical method for picking a random integer: Pick x at random from the half-open interval [0, 1). Then x lies in A_n for a unique integer n. That n is the random integer. ---------------- (( In fact, there is a certain subset A of R/Z such that — for a fixed choice of irrational (say it's sqrt(2)) — the A_n's are each defined as A_n = A + n*sqrt(2) (modulo 1) for all n in Z. )) This means, e.g., that if we defined E = the union of the A_n's for n even and O = the union of the A_n's for n odd then the disjoint union of E and O is R/Z and they are rotated versions of each other. (Similarly if for some K in Z+ we took the union of A_n for all n == j (mod K) for j = 0, 1, ..., K-1.) So there's no assymetry between different integers. —Dan
Yes, you are right, I spoke too soon! On Wed, May 13, 2020 at 5:48 PM Dan Asimov <dasimov@earthlink.net> wrote:
Michael Collins wrote: ----- I think the "symmetry" argument is destroyed by the non-constructive choice of A; i.e. I suspect there are some A which would assign non-zero probability to 37, some which select an even integer with probability other than 1/2 et cetera. -----
No. Regardless of what problems this method may have, asymmetry between different integers is not one of them.
Recap of method: ---------------- Identify the circle R/Z with the half-open interval [0, 1).
Then R/Z is partitioned into subsets
{A_n | n in Z}
whose disjoint union is R/Z and which are all rotated versions of each other.
That allows this theoretical method for picking a random integer:
Pick x at random from the half-open interval [0, 1). Then x lies in A_n for a unique integer n. That n is the random integer. ----------------
(( In fact, there is a certain subset A of R/Z such that — for a fixed choice of irrational (say it's sqrt(2)) — the A_n's are each defined as
A_n = A + n*sqrt(2) (modulo 1)
for all n in Z. ))
This means, e.g., that if we defined
E = the union of the A_n's for n even
and
O = the union of the A_n's for n odd
then the disjoint union of E and O is R/Z and they are rotated versions of each other. (Similarly if for some K in Z+ we took the union of A_n for all n == j (mod K) for j = 0, 1, ..., K-1.)
So there's no assymetry between different integers.
—Dan
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Dan Asimov -
Michael Collins