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 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
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 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:
Theorem 1 Consider 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 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):
Lemma 3 If
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 4 If
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 5 Given 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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