SELF-SIMILAR SETS WITH SUPER-EXPONENTIAL CLOSE CYLINDERS

Baker (2019), Barany and Kaenmaki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method of Baker and obtain further examples of this type. We prove that for any algebraic number beta >= 2 there exist real numbers s, t such that the iterated function system {x/beta, x+1/beta, x+s/beta, x+t/beta} satisfies the above property.