Metric results on inhomogeneously singular vectors

Abstract

Given $\ut\in\Rm$ and any norm $\Vert.\Vert$ on $\Rm$, we consider "inhomogeneously singular" vectors in $\Rm$ that admit an integer vector solution $(q,\underline{p})=(q,p_1,\ldots,p_m)$ to the system \[ 1\leq q\leq Q, \qquad \Vert q\ux-\underline{p}-\underlineθ\Vert\leq cQ^{-1/m} \] for 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 $O(m^{-1})$). We establish slightly weaker bounds for the Hausdorff dimension as well. Our bounds are applicable to the $b$-ary setting, i.e. when restricting to $q$ above integral powers of some given base $b\geq 2$. We further derive similar results for vectors on certain $m$-dimensional fractals, thereby contributing 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 $\Rm$ does not form a comeagre set. We infer this from a general new result that any comeagre set in $\Rm$ has full packing dimension. As another independent consequence of our method, we show that there are subsets $A,B$ of $\mathbb{R}^m$ for which $A+B=B+B=\mathbb{R}^m$ but $A+B$ has Hausdorff dimension less than $m$, and generalizations. 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.