Ubiquitous angles in equiangular sets of lines (Q1580753)

A set of lines through the origin in \(R^d\) is called equiangular if all pairs intersect in the same angle. For a fixed dimension \(d\), there are finitely many nonisometric configurations of \((d+1)\) equiangular lines in \(R^d\) as well as there are finitely many possible angles. It turns out that such angles can occur in only few dimensions or in infinitely many dimensions. The authors prove that there are infinitely many ubiquitous angles \(\theta\) occurring among sets of equiangular lines. Precisely, for all \(d\) large enough, there are \(d+1\) \(\theta\)-equiangular lines in \(R^d\) and there are infinitely many frequent angles, those which occur for finitely many dimensions \(d\), but are not ubiquitous.