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We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ∊(0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The …