Kronecker sequences beyond the torus: nearest-neighbour distances and best returns

Abstract

The classical three-gap theorem says that a finite Kronecker sequence on the circle has at most three gap lengths. We extend this phenomenon to nearest-neighbour distances on quotients $(V\times U)/\Lambda$, where $V$ is a finite-dimensional real normed space, $U$ is an arbitrary ultrametric abelian group and $\Lambda$ is closed. For the quotient metric induced by the maximum product metric, the uniform distance bound is controlled entirely by the real factor. Continuous real directions contained in the subgroup can be factored out; for inner-product metrics, only the real directions generated by the projected subgroup matter. For the maximum norm on $\mathbb{R}^d$ the universal mixed bound is $2^d+1$. These results unify and extend bounds of Chevallier, Haynes--Ramirez, Das--Haynes and Shulga. Best-return denominators behave differently. We show that a recurrence $q_{n+M}\ge q_n+q_{n+1}$ implies at most $M+1$ nearest-neighbour distances in any abelian group with a translation-invariant metric. Shulga's recurrence extends from compact real tori to arbitrary purely real quotients, with the index determined by the rank of the discrete part of the subgroup rather than by the ambient dimension. After adding an ultrametric factor, however, this recurrence can fail, even on an adelic solenoid. Nevertheless a packing recurrence survives for arbitrary mixed quotients, and in effective Euclidean dimensions one and two the one-step loss from the purely real recurrence is sharp.