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

Speaker：Shuhao Zhang (University of Science and Technology of China)

Time：2026-9-19 10:00

Location：Conference Room S106 at Experiment Building at Haiyun Campus

Abstract:

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).