Mathematical Aspects of Analytic Infinite Derivative Gravity Theories

In this note we would like to reiterate in a concise manner several issues of higher and infinite derivative field theories and gravity theories with a hope to wake an interest to solving these questions. Namely, (i) appearance of ghosts even if a ghost-free condition is fixed, (ii) background dependence of the presence of ghosts and possible cure with higher curvature terms, (iii) unexplored connection of the space of solutions in analytic infinite derivative gravity and existence of Fourier transform, and (iv) related problems.