Better Bounds for Planar Sets Avoiding Unit Distances

A 1-avoiding set is a subset of that does not contain pairs of points at distance 1. Let denote the maximum fraction of that can be covered by a measurable 1-avoiding set. We prove two results. First, we show that any 1-avoiding set in that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other) has density strictly less than . For the special case of sets with block structure this proves a conjecture of ErdAs asserting that . Second, we use linear programming and harmonic analysis to show that m(1) (R-2) <= 0.258795.