Self-similar sets with super-exponential close cylinders

S. Baker (2019), B. Bárány and A. 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 S. Baker and obtain further examples of this type. We prove that for any algebraic number $β\ge 2$ there exist real numbers $s, t$ such that the iterated function system $\left \{\frac{x}β, \frac{x+1}β, \frac{x+s}β, \frac{x+t}β\right \}$ satisfies the above property.

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