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 $a_1/q_1 \neq a_2/q_2$ we have $\mathfrak M(q_1,a_1) \cap \mathfrak M(q_2,a_2) \neq \emptyset$.

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert \frac{a_1}{q_1} - \frac{a_2}{q_2} \right\vert\)	\(=\)	\(\displaystyle \left\vert \frac{a_1}{q_1} - \alpha + \alpha - \frac{a_2}{q_2} \right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\leq\)	\(\displaystyle \left\vert \alpha - \frac{a_1}{q_1} \right\vert + \left\vert \alpha - \frac{a_2}{q_2} \right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the Triangle Inequality
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\leq\)	\(\displaystyle 2 \frac QN\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

and

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert \frac{a_1}{q_1} - \frac{a_2}{q_2} \right\vert\)	\(=\)	\(\displaystyle \left\vert \frac{a_1q_2 - a_2q_1}{q_1q_2} \right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\geq\)	\(\displaystyle Q^{-2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This shows that $N \leq 2Q^3 = 2(\log N)^3$.

But by Polynomial Dominates Logarithm, this is impossible for sufficiently large $N$, so the major arcs must be disjoint.

Since the major arcs are pairwise disjoint, closed intervals, by Cover of Interval By Closed Intervals is not Pairwise Disjoint it is not possible that $\mathfrak M = [0,1]$, so $\mathfrak m \neq \emptyset$.

$\Box$