Metrical results on the geometry of best approximations for a linear form, I
Volume 213 / 2024
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 of dimension at least three, Moshchevitin showed that this bound is sharp for $n\ge 2$. We determine the precise value of the Hausdorff and packing dimensions of the set of vectors for which equality occurs, in terms of $n$. Using a result by Moshchevitin, we reduce the problem to determining the Hausdorff and packing dimensions of certain levelsets for the uniform exponent of approximation for a linear form in two variables. The latter metric problem has been completely solved by the recent variational principle by Das, Fishman, Simmons and Urbański, with previous partial results due to Bugeaud, Cheung and Chevallier, which would suffice to prove our Hausdorff dimension formula for $n\ge 4$. The method is generalized to tails of best approximations in higher-dimensional lattices and provides a precise formula for the packing dimensions. However, precise formulas for the Hausdorff dimension of the relevant levelsets for the uniform exponents are not yet known.
