Self-similar sets with super-exponential close cylinders

S. Baker (2019), B. Barany and A. K{a}enm{a}ki (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 S. Baker and obtain further examples of this type. We prove that for any algebraic number $betage 2$ 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.