A note on the points with dense orbit under the expansions of different bases

作者：Wang Wen Ya^*

来源：Chaos Solitons & Fractals, 2015, 77: 235-239.

DOI：10.1016/j.chaos.2015.06.003

It was conjectured by Furstenberg that for any x is an element of[0, 1]\Q,
dim(H) <({2nx(mod 1) : n >= 1})over bar> + dim(H) <({3nx(mod 1) : n >= 1})over bar> >= 1,
where dim(H) denotes the Hausdorff dimension and (A) over bar denotes the closure of a set A. When x is a normal number, the above result holds trivially. In this note, we are aiming at giving explicit non-normal numbers for which the above dimensional formula holds.

• 出版日期2015-8
• 单位华中科技大学