Let A be an m x n matrix and b an m-vector. Twisted Diophantine approximation studies approximations of pairs (A,b) when A is fixed. We present a new geometric idea in metric uniform twisted approximation. Using all successive minima of the diagonal lattice trajectory associated with matrix A, we count lattice points, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted φ-Dirichlet vectors for a large class of functions φ, including all except finitely many power functions. We obtain Hausdorff dimension formulae for twisted singular vectors and prove that taking the intersections causes no dimension drop. We also obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent.
This talk is based on a joint project with Taehyeong Kim; the second part of this project establishes Lebesgue measure zero-one law in asymptotic twisted Diophantine approximation.
