arXiv subjects

Shengyou Wen

Publications and source records attributed to Shengyou Wen.

4 recordsLinked to original sources

arXiv2019

On uniform distribution of $αβ$-orbits

Let $α, β\in (0,1)$ such that at least one of them is irrational. We take a random walk on the real line such that the choice of $α$ and $β$ has equal probability $1/2$. We prove that almost surely the $αβ$-orbit is uniformly distributed module one, and the exponential sums along its orbit has the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension.

math.CA↗

arXiv2015

Remarks on dimensions of Cartesian product sets

Given metric spaces $E$ and $F$, it is well known that $$\dim_HE+\dim_HF\leq\dim_H(E\times F)\leq\dim_HE+\dim_PF,$$ $$\dim_HE+\dim_PF\leq \dim_P(E\times F)\leq\dim_PE+\dim_PF,$$ and $$\underline{\dim}_BE+\overline{\dim}_BF \leq\overline{\dim}_B(E\times F) \leq\overline{\dim}_BE+\overline{\dim}_BF,$$ where $\dim_HE$, $\dim_PE$, $\underline{\dim}_BE$, $\overline{\dim}_BE$ denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of $E$, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product $E\times F$ may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.

math.MG↗

arXiv2015

On thin carpets for doubling measures

We study subsets of $\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Barański carpets are thin for doubling measures.

math.CA↗

arXiv2014

Doubling measures on uniform Cantor sets

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered.

math.MG↗