Random runners are very lonely

Suppose that $k$ runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least $1/k$ from all the other runners. We prove that, with probability tending to one, a much stronger statement holds for random sets in which the bound $1/k$ is replaced by \thinspace $1/2-\varepsilon$. The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs

Correlation among runners and some results on the Lonely Runner Conjecture

The Lonely Runner Conjecture was posed independently by Wills and Cusick and has many applications in different mathematical fields, such as diophantine approximation. This well-known conjecture states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a moment where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a result of Chen on the gap of loneliness and we extend an invisible runner result of Czerwinski and Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing some new results.Comment: 18 page

Rethinking Red Lights: An Economic Approach to Appalachian Prostitution Laws

The legal discourse surrounding prostitution frequently drowns out the voices of sex workers themselves by focusing on questions of morality. But ignoring the voices of those affected by prostitution laws also ignores the driving force behind prostitution: economics. This Note departs from a traditional case study by using interviews with chiefly sex workers and brothel management to craft a more efficient and fair mode of regulating prostitution. By viewing prostitution for what it is—an industry driven by basic economic principles—business law becomes the clear choice for replacing the current, ineffective laws. Furthermore, reshaping prostitution laws to meet the monetary needs of sex workers carries the potential of bridging the wage gap between rural and urban sex workers: a vital step towards financially enabling sex workers in Appalachia