Measuring the spin of Accreting Neutron Stars is important because it can provide constraints on the Equation of State of ultra-dense matter. Particularly crucial to our physical understanding is the discovery of sub-millisecond pulsars, because this will immediately rule out many proposed models for the ground state of dense matter. So far, it has been impossible to accomplish this because, for still unknown reasons, only a small amount of Accreting Neutron Stars exhibit coherent pulsations. An intriguing explanation for the lack of pulsations is that they form only on neutron stars accreting with a very low average mass accretion rate. I have searched pulsations in the faintest persistent X-ray source known to date and I found no evidence for pulsations. The implications for accretion theory are very stringent, clearly showing that our understanding of the pulse formation process is not complete. I discuss which sources are optimal to continue the search of sub-ms pulsars and which are the new constraints that theoretical models need to explain to provide a complete description of these systems
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