6
$\begingroup$

Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive convergents of its simple continued fraction, where $k\ge 1$.

Is it possible that both $|a-(p_k/q_k)|<1/(2q_k^2)$ and $|a-(p_{k+1}/q_{k+1})|<1/(2q_{k+1}^2)$ are true?

I can only prove that at least one of these inequalities is true.

  • 0
    Yes. That is right.2011-03-09

2 Answers 2

5

At least it can happen that not both are true. Example: 333/106 is a convergent to $\pi$, but $(\pi-(333/106)) \cdot 2 \cdot 106^2 \approx 1.87 > 1$.

  • 2
    Of any two consecutive, at least one of them satisfies the condition in the question. That is true. It is an exercise in the Number Theory book by Niven.2011-03-09
1

It is known that $|a-(p_k/q_k)|\le1/(q_kq_{k+1})\le1/(a_{k+1}q_k^2)$ (see, e.g., Hardy and Wright) so if all the partial quotients exceed 2 then all the convergents satisfy your inequalitites.