Very badly approximable numbers

Question

Roth's theorem states that for an algbraic number $a$, $a$ is badly approximated by rationals: for every $\alpha>0$ there is $C>0$ such that for $l\in \mathbb Z$, $$d(la,\mathbb Z)>Cl^{-1-\alpha}.$$

I am wondering if there are some numbers which are even less well approximated: there is $\epsilon(l)$ going to $0$ subpolynomially (typically logarithmically) such that $$d(la,\mathbb Z)>l^{-1}\epsilon(l)?$$

This question is actually related to Well distributed sequence uniformly over small intervals

Answer 1

Yes, indeed one can even take $\epsilon(l)$ to be constant: $d(la,\Bbb Z)\gg_a l^{-1}$ is equivalent to the continued fraction of $a$ having bounded partial quotients. So, for example, every quadratic irrational numbers $a$ has this property.

The reason is that sufficiently good rational approximations to the real number $a$ have to be convergents to $a$. (See for example Theorem 7.14 of Niven/Zuckerman/Montgomery's book.) Moreover, for convergents $h_n/k_n$, the quantity $k_n d(k_n a,\Bbb Z)$ is always between $1/a_{n+1}$ and $1/(a_{n+1}+2)$ (where $a_{n+1}$ is the $(n+1)st$ partial quotient). (See for example problem #6 in Section 7.4 of N/Z/M.)

I believe that when $a=e=2.71828\dots$, the rational approximations satisfy $d(le,\Bbb Z)\gg (l\log l)^{-1}$ (since the continued fraction has a simple pattern with linear upper growth of the partial quotients), which gives another example of the original phenomenon.