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 are super-exponentially close cylinders at all small levels. We adapt the method of obtain further examples this type. prove for any algebraic number \(\beta\ge 2\) real numbers \(s, t\) such system \(\left \{\frac{x}{\beta}, \frac{x+1}{\beta}, \frac{x+s}{\beta}, \frac{x+t}{\beta}\right \}\) satisfies above property.