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Dirichlet's theorem on Diophantine approximation:

For any real number $x$, for integer $N>0$, there exists integers $a$ and $b>0$ with $(a,b)=1$ such that $b\leq N$ and $$|x-\frac a b|<\frac{1}{b(N+1)}.$$

Let $c$ be a fixed positive integer. Is there any version of Dirichlet's theorem to make the following restriction?

version 1) $c|b$

version 2) $(b,c)=1$

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    $\begingroup$ 1) and 2) are contradictory. Did you mean c divides a? $\endgroup$
    – Venkataramana
    Commented Jun 8, 2018 at 0:16
  • $\begingroup$ I mean 1) and 2) are two kinds of restrictions respectively. Obviously 1) and 2) will not exist at the same time... $\endgroup$
    – 7-adic
    Commented Jun 8, 2018 at 3:02

1 Answer 1

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If $p/q$ and $r/s$ are consecutive convergents of the continued fraction for $x$, then $(tp+ur)/(tq+us)$ is a fairly good approximation to $x$, and you may be able to choose small numbers $t$ and $u$ such that $tq+us$ meets your criteria. See Robert Baggenstos, Ein Beitrag zu den diophantischen Approximationen reeller Zahlen, Elem. Math. 60 (2005) 154-170, MR2188009 (2006g:11140).

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