Akademik Veri Yönetim Sistemi
Monatshefte fur Mathematik, cilt.208, sa.1, ss.129-153, 2025 (SCI-Expanded, Scopus)
We continue our study of the geometry of integer best approximations for a linear form in n≥2 real variables. In part I, we determined the Hausdorff and packing dimension of the set of real vectors whose best approximation were ultimately contained in a three-dimensional sublattice of Zn+1 (minimal dimension by results of Jarník, Moshchevitin). Here we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of Zn+1. Our lattices jointly span a lattice of dimension three only, thereby once again verifying that dimension three can be obtained. We determine the exact packing dimension and give an asymptotical formula for the Hausdorff dimension of the according set of real vectors, in dependence of n. Our method combines a new construction for a linear form in two variables n=2 with a result by Moshchevitin to amplify them. To optimize our bounds, we further employ the recent variational principle, as well as estimates for Hausdorff and packing dimensions of Cartesian products and fibers. Our method permits much freedom for the induced classical exponents of approximation.