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5Ultra–Cold Many–Body Systems and Phenomenology of Gravity Theories with Compact DimensionsFoundations of Physics 53 (1): 1-13. 2023.The detection of the number of extra–compact dimensions contained in some gravitational models is analyzed resorting to the discontinuity of the specific heat at the critical temperature of a Bose–Einstein condensate. It is shown that the function relating the number of particles and this discontinuity defines a segment of a straight line whose slope depends upon the number of extra–compact dimensions. The experimental feasibility of the proposal is also considered.