Final review, Mattias Byléhn / Seminar in Algebraic Geometry and Number Theory

Abstract: In analogy with Euclidean and hyperbolic circle problems, we aim to estimate the discrepancy of lattice point counts on asymptotically large balls in regular trees. The lattices we consider are fundamental groups of finite regular graphs X, and we provide upper (and some lower) estimates for the L2-norm of such discrepancies using an analogue of the Fourier transform for regular trees. Sharp upper bounds are obtained from the spectral gap of the adjacency matrix on X, and lower bounds depend significantly on the occurrence of trigonometric eigenvalues, a phenomenon which has no Euclidean or hyperbolic counterpart.