My friend Gugu and I have just uploaded to arXiv our paper .
Remark 1 Currently, this paper deals exclusively with lower bounds on . In its next version, Gugu and I will include upper bounds on .
In what follows, we present a streamlined version of our proof of based on the construction of an explicit Cantor set with .
Remark 2 W e refer to our paper for more refined informations about the structure of near .
1. Perron’s characterization of the classical spectra
Given a bi-infinite sequence and , let
is the usual continued fraction expansion, and
is the th convergent.
In 1921, Perron showed that the classical Lagrange and Markov spectra and are the sets
2. Freiman’s number
In 1973, Freiman showed that
In a similar vein, Theorem 4 in Chapter 3 of Cusick-Flahive book asserts that
for all . In particular, is not isolated in .
Remark 3 As it turns out, is the largest known number in : see page 35 of Cusick-Flahive book.
In what follows, we shall revisit Freiman’s arguments as described in Chapter 3 of Cusick-Flahive book in order to prove the following result:
3. A standard comparison tool
In the sequel, we use the following standard comparison tool for continued fractions is the following lemma (cf. Lemmas 1 and 2 in Chapter 1 of Cusick-Flahive book):
- if and only if ;
Remark 4 For later use, note that Lemma 2 implies that if and for all , then when is odd, and when is even.
4. Proof of Theorem 1
Similarly to the discussions in Cusick-Flahive book, we shall use the next lemma (extracted from Lemma 2 in Chapter 3 of this book):
then where indicates the position in asterisk.
Proof: If (a) occurs, then .
If (b) occurs, then Remark 4 implies that
If (f) occurs, then Remark 4 implies that
We shall also need the following fact:
Proof: See the proof of Theorem 4 in Chapter 3 of Cusick-Flahive book (especially the last paragraph at page 40).
These lemmas allow us to conclude the proof of Theorem 1 along the following lines.
where for all and serves to indicate the zeroth position, then
Proof: On one hand, Remark 4 implies that
and items (a), (b) and (f) of Lemma 3 imply that
for all positions except possibly for with .
On the other hand,
so that for all . This proves the proposition.
is contained in .
5. Lower bounds on
The Gauss map , (where is the fractional part of ) acts on continued fractions as a shift operator:
Therefore, we can use the iterates of the Gauss map to build a bi-Lipschitz map between the Cantor set introduced above and the dynamical Cantor set
Since the Hausdorff dimension is preserved by bi-Lipschitz maps, an immediate corollary of Theorem 1 is:
On the other hand, the Hausdorff dimension was estimated in Subsection 2.2 of this previous post here. In particular, it was shown that:
in the title of this post.