Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then there 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 in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence ${({ϵ}_n)}_{n=1}^{\infty }$ of positive real numbers, there exists an iterated function system ${\{{\varphi }_i\}}_{i\in \mathcal{I}}$ that does not contain exact overlaps and$\min \{|{\varphi }_{\mathbf{a}}(0)-{\varphi }_{\mathbf{b}}(0)|:\mathbf{a},\mathbf{b}\in {\mathcal{I}}^n,\mathbf{a}\ne \mathbf{b},r_{\mathbf{a}}=r_{\mathbf{b}}\}\le {ϵ}_n$ for all $n\in \mathbb{N}$.

分形几何中的几个重要猜想可以归纳如下： $\mathbb{[R}$不等于其期望值，则基础的迭代函数系统包含一个精确的重叠。近年来，在这些猜想方面取得了重大进展。霍奇曼证明，如果Hausdorff维数是一个自相似度量$\mathbb{[R}$不等于期望值，则有一些圆柱体在所有小尺度上都超指数闭合。几年后，Shmerkin证明了${\mathrm{大号}}^q$ 自相似度量的维数 $\mathbb{[R}$。考虑到这些陈述，很自然地想知道是否存在没有精确重叠的迭代函数系统，但是在所有小比例尺上都有超指数接近的圆柱体。在本文中，我们表明确实存在这种迭代功能系统。实际上，我们证明了更多。我们证明对于任何序列${（{ϵ}_{ñ}）}_{ñ=\mathrm{1个}}^{\infty }$ 的正实数存在一个迭代函数系统 ${\{{\varphi }_{\mathrm{一世}}\}}_{\mathrm{一世}\in \mathcal{一世}}$ 不包含确切的重叠$\mathrm{分}\{|{\varphi }_{\mathbf{一种}}（0）-{\varphi }_{\mathbf{b}}（0）|：\mathbf{一种}，\mathbf{b}\in {\mathcal{一世}}^{ñ}，\mathbf{一种}\ne \mathbf{b}，{\mathrm{[R}}_{\mathbf{一种}}={\mathrm{[R}}_{\mathbf{b}}\}\le {ϵ}_{ñ}$ 对全部 $ñ\in \mathbb{ñ}$。