论文标题
类似爆米花的金字塔套件的尺寸
Dimensions of popcorn-like pyramid sets
论文作者
论文摘要
本文涉及一个函数家族的图表的维度理论，这些函数家族包括众所周知的“爆米花函数”及其金字塔样的高维类似物。我们计算这些图的盒子和附加尺寸，以及中间维度，这些尺寸是一个尺寸的家族，在Hausdorff和Box dimension之间插值。作为证据中的工具，我们使用概率理论的Chung $ \ unicode {x2013} $erdős不平等，高维的达芬$ \ unicode {x2013} $ schaeffer类型估计来自Diophantine近似值的估计，以及用于Euler Etient功能的界限。作为应用，我们在图形的分数布朗图像的框尺寸以及不同图之间的Hölder失真上获得边界。
This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung$\unicode{x2013}$Erdős inequality from probability theory, higher-dimensional Duffin$\unicode{x2013}$Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the Hölder distortion between different graphs.
