Counting Bessel zeros and applications

We establish o-minimal definability results and develop point-counting machinery applicable to the zeros of a Bessel function of fixed non-negative real parameter. We also give an Ax-Schanuel theorem for Bessel functions of fixed (complex) parameters and exponential functions. We combine these to prove a sparsity result (weak Manin-Mumford) for points on an algebraic variety whose coordinates are zeros of a Bessel function of fixed (non-negative real) parameter. We apply this result to improve density results on spectral sets for the unit ball motivated by Fuglede's conjecture.