Metric results on inhomogeneously singular vectors

Abstract

We consider vectors in R m that are "inhomogeneously singular", i.e. for some given θ ∈ R m admit an integer vector solution (q, p) = (q, p 1 , . . . , p m ) to the systemfor any c > 0 and all large Q. We show that this set has large packing dimension, and in view of recent deep results by Das, Fishman, Simmons, Urbański, our lower bounds are almost sharp (up to O(m −1 )). Our bound still applies in the b-ary setting, i.e. when restricting to q above integral powers of some given base b ≥ 2. We further establish similar results for vectors that are singular with respect to several θ simultaneously, and further for certain m-dimensional fractals, closely related to a question of Bugeaud, Cheung and Chevallier and complementing recent work by Kleinbock, Moshchevitin and Weiss and by Khalil. Moreover, we show that in contrast to Liouville vectors, the set of singular vectors in R m does not form a comeagre set. We infer this from a general new result that any comeagre set in R m has full packing dimension. The proofs rely on observations on sumsets and a result by Tricot involving Cartesian products and are surprisingly elementary. The topological results further use an observation of Erdős.

Dirichlet spectrum for one linear form

“…The "very singular" vectors induced by very fast decreasing Φ * have frequently occurred in the work of Moshchevitin, particularly regarding geometric aspects of their induced best approximating integer linear forms, we only want to quote [10] exemplarily here. See also [12] for inhomogeneous approximation.…”

Dirichlet spectrum for one linear form

“…The "very singular" vectors induced by very fast decreasing Φ * have frequently occurred in the work of Moshchevitin, particularly regarding geometric aspects of their induced best approximating integer linear forms, we only want to quote [11] exemplarily here. See also [13] for inhomogeneous approximation.…”