Intersections of Cantor sets with hyperbolas and continuous images

Given λ∈(0,1/2), let Cλ=(1−λ)∑i=1∞diλi−1:di∈0,1,∀i≥1be the middle-(1−2λ) Cantor sets with the convex hull [0,1]. We are interested in the set St=(x,y)∈Cλ×Cλ:xy=t, where t∈[0,1]. Since the cases where t=0 or t=1 are trivial, we assume that t∈(0,1) in what follows. We show that if 0.4302≤λ<1/2, the set St has the cardinality of continuum for every t∈(0,1). Besides, we further investigate the continuous image of Cλ×Cλ under the function fk(x,y)=xky, that is, for any given 2≤k∈N, we give a sufficient condition for set xky:x,y∈Cλ to be the interval [0,1]. Our observations reveal that the behavior exhibited by the image of Cλ×Cλ under the function fk(x,y)=xky is complex and depends on the parameters k and λ.