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Arithmetic on self-similar sets
Indagationes Mathematicae ( IF 0.846 ) Pub Date : 2020-05-22 , DOI: 10.1016/j.indag.2020.05.003
Bing Zhao; Xiaomin Ren; Jiali Zhu; Kan Jiang

Let K1 and K2 be two one-dimensional homogeneous self-similar sets with the same ratio of contractions. Let f be a continuous function defined on an open set U⊂R2. Denote the continuous image of f by fU(K1,K2)={f(x,y):(x,y)∈(K1×K2)∩U}.In this paper we give a sufficient condition which guarantees that fU(K1,K2) contains some interiors. Our result is different from Simon and Taylor’s Simon and Taylor (2020, Proposition 2.9) as we do not need the condition that the product of the thickness of K1 and K2 is strictly greater than 1. As a consequence, we give an application to the univoque sets in the setting of q-expansions.
更新日期:2020-05-22

 

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