Almost periodic sets and measures on the torus

We establish some “number theoretical” results about a continuous functionh from the circleT into itself, which generalize Kronecker’s theorem in several ways. These results are used to characterize the almost periodic sets of the flow on the torusT ² generated by (θ, φ) → (θ+α, φ+h(θ)), where α is irrational. The almost periodic measures are characterized in the caseh(θ)=θ.