My friend Gugu and I have just uploaded to arXiv our paper .

In this article, we study the complement of the Lagrange spectrum in the Markov spectrum near a non-isolated point found by Freiman, and, as a by-product, we prove that its Hausdorff dimension is

Remark 1Currently, 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 2W 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

Here,

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

where

**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 3As 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:

Theorem 1Consider the alphabet consisting of the words and . Then,

**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 4For 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):

Lemma 3If contains any of the subsequences

- (a)
- (b)
- (f)

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:

Lemma 4If is a bi-infinite sequence such that

then .

*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.

Proposition 5Given a bi-infinite sequence

where for all and serves to indicate the zeroth position, then

*Proof:* On one hand, Remark 4 implies that

and

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.

At this point, the proof of Theorem 1 is complete: in fact, Proposition 5 and Lemma 4 together imply that

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:

By putting Corollary 6 and Proposition 7, we conclude the desired estimate

in the title of this post.

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