We will prove a conjecture [1] on upper bound of asymptotic gap in dimension in unitary compact 2D CFT, proving it to be 1 using a "extremal" function [2]. The notion of asymptotic gap can be generalized to (h, \bar{h}) plane [3]. We will discuss how the extremal function used in [2] can be put into a bigger (natural) framework of an extremal problem of entire function with certain exponential growth [4]. This facilitates a generalization of construction of optimal function (within the class of Bandlimited function) and hence the optimal gap on (h, \bar{h}) plane. 

Reading material:

[1] Baur Mukhametzhanov, Alexander Zhiboedov, Modular Invariance, Tauberian Theorems, and Microcanonical Entropy, arXiv: 1904.06359 [hep-th]

[2] Shouvik Ganguly, SP, Bounds on density of states and spectral gap in CFT_2, arXiv: 1905.12636 [hep-th] (in particular section 4)

[3] SP, Zhengdi Sun, Tauberian-Cardy formula with spin, arXiv: 1910.07727 [hep-th]
[4] D. V. Gorbachev,  Extremum Problems for Entire Functions of Exponential Spherical Type, Mathematical Notes, Vol 68, N0. 2, 2000

I will mostly be focussing on the extremal problem. More reading material on application of Tauberian theorem can be found in:
1. D. Pappadopulo, S. Rychkov, J. Espin, and R. Rattazzi, Operator product expansion convergence in conformal field theory, Physical Review D 86 no. 10, (2012) 105043.

2. J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119, arXiv:1709.00008 [hep-th].

3. B. Mukhametzhanov and A. Zhiboedov, Analytic euclidean bootstrap, arXiv preprint arXiv:1808.03212 (2018).

4. SP, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, arXiv:1906.11223 [hep-th].

5. Appendix C of D. Das, S. Datta, and SP, Charged structure constants from modularity, JHEP 11 (2017) 183, arXiv:1706.04612 [hep-th].