3 ms·
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 t
by 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 agoYeah 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 agoRead 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 agoThe 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!