Unique expansions and intersections of Cantor sets

作者：Baker, Simon^*; Kong, Derong

来源：Nonlinearity, 2017, 30(4): 1497-1512.

DOI：10.1088/1361-6544/aa6078

• expansions in non-integer bases
• unique expansions
• Hausdorff dimension
• intersections of Cantor sets
• digit frequencies

To each epsilon(1/3,1/2) we associate the Cantor set T-alpha={Sigma infinity i=1 epsilon(i)epsilon{0,1},i >= 1}. @@@ In this paper we consider the intersection T alpha boolean AND(T alpha+t) T epsilon Rfor any translation t?R. We pay special attention to those t with a unique {-1,0,1} alpha-expansion, and study the set @@@ D alpha:={dim(H)(T alpha boolean AND(T alpha+t)):t has a unique {-1,0,1}alpha-expansion}. We prove that there exists a transcendental number alpha KL approximate to 0.39433 such that: D alpha is finite for alpha epsilon(alpha KL,1/2), D-alpha KL is infinitely countable, and D alpha contains an interval for alpha epsilon(1/3,alpha KL). We also prove that D-alpha equals [0,log2/log alpha] if and only if alpha epsilon(1/3,3-root 5/2]. As a consequence of our investigation we prove some results on the possible values of dim(H)(T alpha boolean AND(T alpha+t)) when T alpha boolean AND(T alpha+t) is a self-similar set. We also give examples of t with a continuum of {-1,0,1} ?-expansions for which we can explicitly calculate dimHT alpha boolean AND(T alpha+t)), and for which T alpha boolean AND(T alpha+t) is a self-similar set. We also construct ? and t for whichT alpha boolean AND(T alpha+t) contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those t with a unique {-1,0,1} alpha-expansion.

• 出版日期2017-4
• 单位扬州大学