ARPAVAX:reeds (07/27/82)
In reference to dmr's 3 questions: Maybe I misunderstand two of the three
problems, but they sound easy.
Q2) (Not sure whose this is) A grasshopper and a dime inhabit a plane.
The grasshopper is able to choose any distance to jump. However, with
each jump the angle it takes is random and uniform. Does the grasshopper
have a strategy that will cause it to land eventually on the dime, no
matter how far away it starts? ("eventually" means "with probability 1").
Sure. If the grasshopper always makes hops of unit length but at randomly
chosen angles (uniformly distributed on 0 to 2*pi, the angles of different
hops being chosen independently of each other) then for every finite area
compact K in the plane the following happens with probablity 1: the grass-
hopper makes infinitely many visits to K. Not quite sure of all the details
at the moment but basically use the fact that in dimensions 1 & 2 finite
variance random walks are recurrent, and that the probability law for the
grasshopper's position after n > 1 hops is continuous (with respect to Lebesgue
measure).
Q3) (Bill Gosper) A flea starts out at the origin on a plane.
It chooses an angle theta. Then it makes a series of jumps of unit
length, the first at angle theta, ... , the n-th at angle n*theta...,
all measured w.r.t. the line from the flea's current position towards
x = +infinity.
For what values of theta will the flea's path lie within a bounded distance
of the origin?
This sounds even easier. Use complex numbers. The n-th jump is exp(i*n*theta)
= cos(n*theta) + i*sin(n*theta). The position after n jumps is the sum
of exp(i*k*theta) over all k=1,2,...,n, which is
(exp(i*(n+1)*theta) - exp(i*theta)) / (exp(i*theta) - 1)
for all values of theta != 0, by summing the (finite) geometric series. So
what happen to this as n -> infinity? N enters the above formula only in
exp(i*(n+1)*theta), which is bounded as n -> infinity. The flea does not travel
far.
If theta == 0 all hops are due East, and only in this case will the flea travel
far. (Off into the rising sunset?)