12 ms·

Isn't it true that if the runners' speeds are all real numbers which are not rational multiples of each other, eventually you'll see every distribution of posit

Isn't it true that if the runners' speeds are all real numbers which are not rational multiples of each other, eventually you'll see every distribution of positions of the runners, or at least come arbitrarily close?

tromp 4y ago
That seems true, by induction on the number of runners. But the conjecture does allow for rational speed ratios.
- deleted 4y ago
  [deleted]
jameshart 4y ago
The challenge isn't to find sets of numbers that guarantee loneliness will occur- as you say, arbitrary reals that aren't rational multiples should accomplish that handily. The conjecture isn't that such sets can be found - it's much stronger: that if the runners have any set of distinct speeds, then they all experience loneliness. And, according to the article, and perhaps surprisingly: > The conjecture can be reduced to restricting the runners' speeds to positive integers: If the conjecture is true for n runners with integer speeds, it is true for n numbers (sic) with real speeds. i.e., the conjecture is that n runners with speeds which are all rational multiples of one another will experience loneliness.
- gweinberg 4y ago
  Yeah sure I understand that. Just wanted to verify that at the offset we can elimate an uncountable number of possibilities and so for each n there are only a countable number of cases that need to be considered.
  - eru 4y ago
    Read on in the article. Someone even reduced it further, so you only need to check a finite number of cases for each fixed n.
- karmakaze 4y ago
  The conjecture is stronger: real != rational > This convention is used for the rest of this article. Wills' conjecture was part of his work in Diophantine approximation, the study of how closely fractions can approximate irrational numbers. I couldn't have imagined that we might make a connection between integers and irrational/real numbers.
- JadeNB 4y ago
  > > The conjecture can be reduced to restricting the runners' speeds to positive integers: If the conjecture is true for n runners with integer speeds, it is true for n numbers (sic) with real speeds. No need for sic when quoting Wikipedia; you can (and I have) just fix it!
impendia 4y ago
I believe that if the runners speeds are all real numbers which are linearly independent over the rationals, then the associated dynamical system will be ergodic https://en.wikipedia.org/wiki/Ergodicity https://en.wikipedia.org/wiki/Ergodicity and what you say is true. But in addition to your example you have to worry about, e.g. sqrt(2), sqrt(3), and sqrt(2) + sqrt(3).
- contravariant 4y ago
  I agree that it's probably ergodic, but to show every distribution will definitely come about (approximately) you'll need a slightly stronger property than just ergodicity. Ergodicity guarantees it is almost always the case, but if you want it to always be the case you'll need it to be uniquely ergodic. Since irrational rotations are uniquely ergodic I reckon this system will also be, but I think the proof won't be trivial.

tromp 4y ago

That seems true, by induction on the number of runners. But the conjecture does allow for rational speed ratios.

deleted 4y ago
[deleted]

jameshart 4y ago

The challenge isn't to find sets of numbers that guarantee loneliness will occur- as you say, arbitrary reals that aren't rational multiples should accomplish that handily. The conjecture isn't that such sets can be found - it's much stronger: that if the runners have any set of distinct speeds, then they all experience loneliness. And, according to the article, and perhaps surprisingly: > The conjecture can be reduced to restricting the runners' speeds to positive integers: If the conjecture is true for n runners with integer speeds, it is true for n numbers (sic) with real speeds. i.e., the conjecture is that n runners with speeds which are all rational multiples of one another will experience loneliness.

gweinberg 4y ago
Yeah sure I understand that. Just wanted to verify that at the offset we can elimate an uncountable number of possibilities and so for each n there are only a countable number of cases that need to be considered.
- eru 4y ago
  Read on in the article. Someone even reduced it further, so you only need to check a finite number of cases for each fixed n.
karmakaze 4y ago
The conjecture is stronger: real != rational > This convention is used for the rest of this article. Wills' conjecture was part of his work in Diophantine approximation, the study of how closely fractions can approximate irrational numbers. I couldn't have imagined that we might make a connection between integers and irrational/real numbers.
JadeNB 4y ago
> > The conjecture can be reduced to restricting the runners' speeds to positive integers: If the conjecture is true for n runners with integer speeds, it is true for n numbers (sic) with real speeds. No need for sic when quoting Wikipedia; you can (and I have) just fix it!

impendia 4y ago

I believe that if the runners speeds are all real numbers which are linearly independent over the rationals, then the associated dynamical system will be ergodic https://en.wikipedia.org/wiki/Ergodicity https://en.wikipedia.org/wiki/Ergodicity and what you say is true. But in addition to your example you have to worry about, e.g. sqrt(2), sqrt(3), and sqrt(2) + sqrt(3).

contravariant 4y ago
I agree that it's probably ergodic, but to show every distribution will definitely come about (approximately) you'll need a slightly stronger property than just ergodicity. Ergodicity guarantees it is almost always the case, but if you want it to always be the case you'll need it to be uniquely ergodic. Since irrational rotations are uniquely ergodic I reckon this system will also be, but I think the proof won't be trivial.