Relationship between Dirichlet's Approximation Theorem and Convergents For any real number $r$, the convergents to the continued fraction expansion of $r$ satisfy Dirichlet's approximation inequality of $|r - \frac{p}{q}| < \frac{1}{q^2}$.
Does this go the other way? That is, if given some rational number $p/q$, we have that $|r - \frac{p}{q}| < \frac{1}{q^2}$, does that necessarily mean that it is a convergent of the continued fraction expansion of $r$?
It does seem, from this page, that we have a similar result that the convergents, and only the convergents, minimize $|r - \frac{p}{q}| \cdot q$ among all smaller rationals, so we would need to show that whenever this happens, we also have $|r - \frac{p}{q}| \cdot q < \frac{1}{q}$.
EDIT: Legendre has also proven that if $|r - \frac{p}{q}| < \frac{1}{2q^2}$ then $p/q$ is a convergent of $r$. So the main question is what happens if for whatever $p/q$ we have that $\frac{1}{2q} < |r - \frac{p}{q}| < \frac{1}{q}$: can we determine that $p/q$ is a convergent, or a semiconvergent, or anything at all?
 A: The answer above is great; I just wanted to summarize in some more succinct terms:

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*The answer to my original question is no. There can be "false positives" for which $|r - p/q| < 1/q^2$, but which are not convergents. One example is, if we're trying to approximate $\log_2(3/2)$, that $10/17$ has the required property but is only a semiconvergent. So this is necessary but not sufficient.


*I also talked about how if $|r - p/q| < 1/(2q^2)$, then we know we have a convergent. This also does not go the other way, because now there can be "false negatives." Another good example is $24/41$, which is also convergent of $\log_2(3/2)$ but which doesn't have the required property. So this is sufficient but not necessary, and now we get "false negatives" if we are looking for convergents.


*There is an important result of Fatou which basically says, paraphrasing slightly, that if we do have $|r - p/q| < 1/(2q^2)$, which again is sufficient but not necessary, the "extra" rationals satisfying this property which are not convergents are all semiconvergents that are directly adjacent to some convergent.


*I was hoping for some magic $k$ such that if the error is less than $1/(kq^2)$, that's necessary and sufficient to make it a convergent. But, it appears that non-convergents can get arbitrarily close to the $1/(2q^2)$ bound, and going the other way, convergents can get arbitrarily close to the $1/(q^2)$ bound.
