Abstract. We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of R d to corners in subsets of

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to zero sufficiently slowly contains a 3-term term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of 3-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.Date: September 15, 2017.

. The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit‐length circular track, consider m runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least 1/m from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.