Density estimates of 1-avoiding sets via higher order correlations (Q2136845)

The paper is concerned with the maximum upper density of a Lebesgue measurable planar point set that does not contain two points at distance 1 apart. This old problem has received much interest recently. Paul Erdős conjectured that the maximum upper density is at most \(1/4\), while the best known lower bound is about 0.22936. The paper under review sets a new upper bound, 0.25442, using a combination of Fourier analytic and linear programming methods, getting tantalizingly close to the conjecture. One of the new techniques is studying the upper density of the intersection of 3 translates of a Lebesgue measurable planar point set that does not contain two points at distance 1 apart.