We consider the universality of existence and saturation of asymptotic bounds in various quantities in 2D CFT. In particular, we focus on previously derived upper and lower bounds on the number of operators in a window of scaling dimensions [\Delta−\delta,\Delta+\delta] at asymptotically large \Delta in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval [\Delta−\delta ,\Delta+\delta] and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. When the width of the interval is integer, the bounds are saturated by known partition functions with integer-spaced spectra. We further show with numerical assistance that one can see morally similar bounds and saturation in asymptotics of various OPE coefficients and two point correlator in heavy states. We comment on how these shed light on our understanding of chaotic vs non-chaotic CFT.
