Universal and Simultaneous Fractal Transference for Subsets of N^d via Continued Fractions

报告人：张书豪（中国科学技术大学）

时  间：2026年9月19日10:00

地  点：海韵园实验楼S106

内容摘要:

In this talk, I will present joint work with Zhuowen Guo, Kangbo Ouyang, and Jiahao Qiu. We establish universal and simultaneous fractal transference principles that transfer quantitative properties of largeness for subsets of  N^d to Hausdorff-dimensional properties of subsets of [0,1)^d through continued-fraction expansions. Our results extend the one-dimensional transference theorem of Nakajima--Takahasi, to higher dimensions and strengthen it already in dimension one. We prove that upper density admits a universal realization of the maximal Hausdorff dimension d/2, while upper density and upper Banach density can be preserved simultaneously for any prescribed countable family of subsets of N^d; in dimension one, both admit a single universal realization. We further establish a relative transference principle for uniformly balanced sets with polynomial density exponent α, obtaining the exact Hausdorff dimension d/(2α).

个人简介：

张书豪，中国科学技术大学数学科学学院博士研究生，主要从事拓扑动力系统与遍历理论研究。研究兴趣包括动力系统中的独立性、熵、回复性质与遍历平均，以及遍历理论与组合数论、分形几何等方向的交叉问题。最近，与合作者共同完成了Owings 和集问题的证明。

联系人：连政星