The lonely runner conjecture

Weaker versions have been proven, even for higher dimensions.

Is it just me? I really hate how the problem is phrased on that page. Here's the original:

"33 horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can pass one another. Can they ride in this fashion for an arbitrarily long time?"

Here's a quick take on how I would rephrase it:

"33 horsemen are riding in the same direction along a circular road. There is just a single point on the road where any number of horsemen can pass each another; everywhere else, a faster rider coming up behind a slower rider will be stuck and have to slow down. Is there a set of pairwise distinct speeds the horsemen can ride at such that the riders can all start at the passing point, ride constantly at their given speed for an arbitrarily long time, and only meet each other at the passing point (thus never having to change speed)?"

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

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.

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

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).

The pictures are just similar.

You don't have any smooth movement in consistent hashing. And you can pick your inputs carefully, instead of worrying about the worst case like in the linked conjecture.

Btw, you might like https://en.wikipedia.org/wiki/Rendezvous_hashing

n-hypercubes are very relevant and tangible. they model the state of your computer, for one.

Mathematicians are generally interested in the implications to other bits of math.

A hypercube is just the higher-dimensional analog of a cube. A 2-hypercube is a square, a 3-hypercube is a cube, and so on.

Edit: I know complaining about downvotes is against the rules, and I’m not complaining, because I really don’t care about my karma on this site — but I am really confused and curious how people could possibly be opposed to me explaining a mathematical concept to someone who expressed frustration at not knowing it.