Vinogradov's Theorem/Lemma 1

Lemma
For sufficiently large $N$ the major arcs are pairwise disjoint, and the minor arcs are non-empty.

Proof
Suppose that for some admissible $a_1/q_1 \ne a_2/q_2$ we have:
 * $\mathcal M \left({q_1, a_1}\right) \cap \mathcal M \left({q_2, a_2}\right) \ne \varnothing$

Then using the definition of the major arcs, for $\alpha$ in the intersection we have:

and

This shows that:
 * $N \le 2Q^3 = 2 \left({\log N}\right)^3$

But by Polynomial Dominates Logarithm, this is impossible for sufficiently large $N$.

Therefore the major arcs must be disjoint.

We have that the major arcs are pairwise disjoint closed intervals.

So by Cover of Interval By Closed Intervals is not Pairwise Disjoint it is not possible that $\mathcal M = \left[{0 \,.\,.\, 1}\right]$.

So it follows that:
 * $\mathcal m \ne \varnothing$