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] [ Δ − δ , Δ + δ ] at asymptotically large Δ 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] [ 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 ≥ 0 δ ≥ 0 . When 2δ ∈ Z≥ 0 2 δ ∈ ℤ ≥ 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 c > 1 theories.