On angles between linear subspaces in \(\mathbb{R}^4\) and the singularity (Q6569614)

Denote \(\| \cdot \|\) the Euclidean norm in \(\mathbb R^4\). Let \(\psi(X,Y)=\| X\wedge Y\|/(\| X\|\cdot \| Y\|)\) be the angle of vectors \(X,Y\in\mathbb R^4\). A two-dimensional subspace \(B\) of \(\mathbb R^4\) is rational if it has a basis of vectors with rational components. Denote the set of rational subspaces by \(\mathfrak R_4(2)\). Let us call by the height \(H(B)\) of a subspace \(B\) the volume of the fundamental domain of the two-dimensional lattice \(B\cap\mathbb Z^4\) contained in it. For a two-dimensional subspaces \(A\) and \(B\) set \(\psi_1(A,B)=\min_{X\in A\setminus\{ 0\}, Y\in B\setminus\{ 0\}} \psi(X,Y)\). Let \(f(t)\) be an arbitrary strictly positive decreasing function on \([1,\infty)\). Then the author proves that there exists subspace \(C\) such that for every \(t\geq t_0\) we have \(0<\min_{D\in\mathfrak R_4(2), H(D)\leq t} \psi_1(C,D)<f(t)\).