Iterated function systems with super-exponentially close cylinders

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