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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 $[Δ- δ,Δ+ δ]$ at asymptotically large $Δ$ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[Δ- δ,Δ+ δ]$ 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 $δ\geq 0$. When $2δ\in \mathbb Z_{\geq 0}$ 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.