Beurling-Selberg Extremization and Modular Bootstrap at High Energies

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\mathrm{\Delta }-\delta ,\mathrm{\Delta }+\delta ]$ at asymptotically large $\mathrm{\Delta }$ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[\mathrm{\Delta }-\delta ,\mathrm{\Delta }+\delta ]$ 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. We review solutions of this problem and present the corresponding bounds on the number of operators for any $\delta \geq 0$\mathrm{.}When$2\delta \in \mathbb Z_{\geq 0}$ the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in $c>1$ theories.