Using Minkowski's Theorem to prove existence.

Question

I need to use Minkowski’s theorem to show that if α ∈ R and Q ∈ N then there exist q, a ∈ Z such that 1 ≤ q ≤ Q and |qα − a| ≤ 1/Q. I believe that I need to define the lattice in R: Λ = {qα : 0<α<1/Q} but am not sure where to go from there. Is this the correct lattice to use and if so how would I use it in the proof? Any help would be greatly appreciated. Thanks.

Answer 1

Consider the set of the form $ S = \{(x,y) \in \mathbb{R}^2 : |\alpha x - y|\leq Q^{-1} , |x|< Q\}$ and the lattice $\Lambda$ spanned by $\begin{pmatrix} 1 \\ 0\end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0\end{pmatrix}$. You can calculate that $V(S) = 4$ and $2^2\det(\Lambda) = 4$ therefore since $S$ is symmetric, convex and compact by the strong minkowski property you can show that there exists a lattice point in $S$. This lattice point consists of $(q,a)\in \mathbb{Z}^2$ such that $|q\alpha - a|\leq Q^{-1} , |q|< Q$ as required. Since S is symmetric if $-Q \leq q \leq -1$ we know that the point $(-q,-s)$ also satisfies your equation with the additional caveat that $ 1 \leq q \leq Q$ as required. P.S. MA 257 gets pretty techy :)