Diophantine direction

A vector $v\in\mathbb R^n$ is a Diophantine direction with constants $c>0$ and $\tau\ge0$ if

|v\cdot m|\ge c(1+|m|)^{-\tau} \qquad(m\in\mathbb Z^n\setminus\{0\}).

The dot product and norm are Euclidean. This convention is a lower bound on distance to zero, not on distance to the nearest integer; the latter appears in a different Diophantine condition for rotations.

Example and dependence

For a quadratic irrational $\alpha$ , the vector $(1,\alpha)$ satisfies the condition with $\tau=1$ by the quadratic lower bound. Multiplying a direction by a nonzero scalar rescales $c$ . Estimates based on the condition must track their dependence on $c$ and $\tau$ ; the bound excludes every nonzero resonant Fourier mode.