Self-similar sets with super-exponential close cylinders

Abstract

Baker (2019), Bárány and Käenmäki (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 \(\beta\ge 2\) there exist real numbers \(s, t\) such that the iterated function system \(\left \{\frac{x}{\beta}, \frac{x+1}{\beta}, \frac{x+s}{\beta}, \frac{x+t}{\beta}\right \}\) satisfies the above property.