Metrical results on the geometry of best approximations for a linear form

TL;DR

This paper investigates the geometric properties of best approximations for linear forms in multiple variables, determining the dimensions of specific sets and providing new constructive proofs and methods.

Contribution

It precisely calculates Hausdorff and packing dimensions of sets where equalities occur and introduces a new construction for linear forms in two variables.

Findings

Exact Hausdorff and packing dimensions of the set where equality occurs.

Existence of vectors with best approximations in a union of two 2D sublattices.

Alternative proof of Moshchevitin's result using new constructions.

Abstract

Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff and packing dimension of the set where equality occurs, in terms of $n$. Moreover, independently we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of $\Z^{n+1}$. Our lattices jointly span a lattice of dimension three only, thereby leading to an alternative constructive proof of Moshchevitin's result. We determine the packing dimension and up to a small error term $O(n^{-1})$ also the Hausdorff dimension of the according set. Our method combines a new construction for a linear form in two variables $n=2$ with a result by Moshchevitin to amplify them. We further employ the recent variatonal principle and some of its consequences, 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.