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Several important conjectures in Fractal Geometry can be summarised asfollows: If the dimension of a self-similar measure in $\mathbb{R}$ does notequal its expected value, then the underlying iterated function system containsan exact overlap. In recent years significant progress has been made towardsthese conjectures. Hochman proved that if the Hausdorff dimension of aself-similar measure in $\mathbb{R}$ does not equal its expected value, thenthere are cylinders which are super-exponentially close at all small scales.Several years later, Shmerkin proved an analogous statement for the $L^q$dimension of self-similar measures in $\mathbb{R}$. With these statements inmind, it is natural to wonder whether there exist iterated function systemsthat do not contain exact overlaps, yet there are cylinders which aresuper-exponentially close at all small scales. In this paper we show that suchiterated function systems do exist. In fact we prove much more. We prove thatfor any sequence $(\epsilon_n)_{n=1}^{\infty}$ of positive real numbers, thereexists an iterated function system $\{\phi_i\}_{i\in \mathcal{I}}$ that doesnot contain exact overlaps and$$\min\left\{|\phi_{\mathbf{a}}(0)-\phi_{\mathbf{b}}(0)|:\mathbf{a},\mathbf{b}\in \mathcal{I}^n,\, \mathbf{a}\neq \mathbf{b},\,r_{\mathbf{a}}=r_{\mathbf{b}}\right\}\leq \epsilon_n$$ for all $n\in\mathbb{N}.$

Iterated function systems with super-exponentially close cylinders