A technical note on the arithmetic cone of smooth periodic vector fields

For a smooth periodic vector field on the $n$-torus, the contraction method for establishing strong rotation vectors extends only to those asymptotic directions $\rho \in \R^n$ for which a certain spectral sum remains uniformly bounded along a sequence of rational approximations. We introduce the "arithmetic cone" $\mathfrak{C}(f)$, defined as the set of all $\rho$ admitting such an approximation. We establish its basic algebraic property: it is a cone. We prove that, under a uniform contraction condition, every element of $\mathfrak{C}(f)$ yields a strong rotation vector for the dynamics. The construction reveals a precise link between the Fourier asymptotics of $f$ and the arithmetic of admissible rotation directions. In the second part, we introduce the class of "spectrally admissible" fields $\mathcal{A}_{ spec }$, for which the cone of the augmented field equals the whole space, and we show that it contains all finite trigonometric polynomials.