I am reading about Weyl differencing for estimating exponential sums and there is a line that I don't quite understand.
We have a real number $\alpha$ and using Dirichlets approximation theorem we can find coprime integers $a,q$ and another integer $N$ with $1 \leq q \leq 2N$ such that $$\big{|}2\alpha - \frac{a}{q} \big{|} \leq \frac{1}{2Nq}. $$ Thats all fine by me.
They then state for any $\ell$ with $1 \leq \ell < N$ and $\ell \not\equiv 0 \:(\text{mod}\: q)$ we have $$||2\alpha \ell|| \geq \frac{1}{2}||a\ell/q|| $$ where $||\cdot ||$ denotes distance to the nearest integer. I cant quite figure this line out. So I know the distance between the two real numbers is of course less than a half, but how can we get things so precise. What if $2\alpha \ell$ was super close to an integer - I cant see why $a\ell/q$ would also have to be.
Write $r$ for the residue of $l$ mod $q$. From your hypothesis you have modulo 1 \[ 2\alpha l=ar/q+\theta \] where $|\theta |\leq 1/2q$. If $\theta $ is this small then $ar/q+\theta $ must lie in \[ \left (\frac {ar-1}{2q},\frac {ar+1}{2q}\right )\] and so is obviously bigger than $||ar/2q||$, unless $r=0$.
Your last comment: $q\not |l$ rules out $al/q$ an integer, and the distance condition makes sure $\alpha l$ still stays far enough away from an integer (stays close to some $r/q$).