Re: [math-fun] Fwd: HAKMEM 32
What does "have a cluster point at 0" mean here? E.g., it might mean that the underlying density is infinite, or has a local max, at 0. --Dan << ITEM 32 (Schroeppel) Take a random real number and raise it to large powers; we expect the fraction part to be uniformly distributed. . . . . . . Now, 3 + sqrt2 is suspicious; it looks non-uniform, and seems to have a cluster point at zero. PROBLEM: Is it non-uniform?
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Somewhat similar to the situation with sqrt2: A combination of a uniform distribution, and a spike of points near 0. I think sqrt2+sqrt3 will behave this way: The odd powers looking uniform (mod 1), while the even powers cluster near 0. Rich --------------- Quoting Dan Asimov <dasimov@earthlink.net>:
What does "have a cluster point at 0" mean here?
E.g., it might mean that the underlying density is infinite, or has a local max, at 0.
--Dan
<< ITEM 32 (Schroeppel) Take a random real number and raise it to large powers; we expect the fraction part to be uniformly distributed. . . . . . . Now, 3 + sqrt2 is suspicious; it looks non-uniform, and seems to have a cluster point at zero. PROBLEM: Is it non-uniform?
I think sqrt2+sqrt3 will behave this way: The odd powers looking uniform (mod 1), while the even powers cluster near 0.
Right, because (sqrt2+sqrt3)^2n+(sqrt2+sqrt3)^-2n is always integral for integral n. This is related to Rubik's Cube, but this textarea is too small.
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Dan Asimov -
rcs＠xmission.com -
Tom Rokicki