论文标题
对不均匀奇异向量的度量结果
Metric results on inhomogeneously singular vectors
论文作者
论文摘要
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 \[ 1\leq q\leq Q, \ qquad \ vert q \ ux- \下划线{p} - \lunderlineθ\ vert \ leq cq^{ - 1/m} \]对于任何$ c> 0 $和所有大$ q $。我们证明，这套集合具有较大的包装维度，并且鉴于DAS，Fishman，Simmons，Urbański最近的深层结果，我们的下限几乎是锋利的（最高$ O（M^{ - 1}）$）。我们也为Hausdorff维度建立了较弱的界限。我们的界限适用于$ b $ - ary设置，即限制在某些给定基础$ b \ geq 2 $的积分功率上方的$ q $时。我们在某些$ m $维的分形上进一步取得了类似的结果，从而导致了Bugeaud，Cheung和Chevallier的问题，并补充了Kleinbock，Moshchevitin和Weiss和Weiss以及Khalil的最新工作。此外，我们表明，与liouville矢量相比，$ \ rm $中的单数矢量集并不形成稳定的套件。我们从一般的新结果中推断出来，任何comeagre设置为$ \ rm $都具有完整的包装维度。作为我们方法的另一个独立后果，我们表明有$ \ mathbb {r}^m $的子集$ a，b $ a+a+a+b = b = b+b = \ mathbb {r}^m $，但$ a+a+a+a+a+b $的hausdorff dimension dimension dimension dimension dimension by $ m $和概括。这些证明依赖于对总和的观察以及涉及笛卡尔产品的Tricot的结果，并且令人惊讶的是基本。拓扑结果进一步使用了ERDS的观察。
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.
