Posted by: matheuscmss | March 10, 2018

Cusick’s conjecture on Lagrange and Markov spectra after 3.46…

The collection of best constants ${c}$ for the Diophantine approximation problem of finding infinitely many rational solutions ${p/q\in\mathbb{Q}}$ to the inequality

$\displaystyle |\alpha-\frac{p}{q}|<\frac{1}{cq^2}$

with ${\alpha\in\mathbb{R}\setminus\mathbb{Q}}$ is encoded by the so-called Lagrange spectrum ${L}$.

In a similar vein, the Markov spectrum ${M}$ encodes best constants for a Diophantine problem involving indefinite binary quadratic real forms.

These spectra were first studied in a systematic way by A. Markov in 1880, and, since then, their structures attracted the attention of several mathematicians (including Hurwitz, Perron, etc.).

Among the basic properties of these spectra, it is worth mentioning that ${L\subset M}$ are closed subsets of the real line. Moreover, the works of Markov from 1880 and Hall from 1947 imply that

$\displaystyle L\cap(-\infty, 3) = M\cap(-\infty, 3) = \{\sqrt{5}<\sqrt{8}<\dots\}$

is a increasing sequence of quadratic surds converging to ${3}$, and

$\displaystyle L\cap[6,\infty) = M\cap[6,\infty) = [6,\infty)$

On the other hand, it took some time to decide whether ${L=M}$. Indeed, Freiman proved in 1968 that ${M\setminus L\neq\emptyset}$ by exhibiting a countable (infinite) collection of isolated points in ${M\setminus L}$. After that, Freiman constructed in 1973 an element of ${M\setminus L}$ which was shown to be a non-isolated point of ${M\setminus L}$ by Flahive in 1977.

A common feature of these examples of elements in ${M\setminus L}$ is the fact that they occur before ${\sqrt{12}=3.46\dots}$ In 1975, Cusick conjectured that there were no elements in ${M\setminus L}$ beyond ${\sqrt{12}}$.

In our preprint uploaded to arXiv a couple of days ago, Gugu and I provide the following negative answer to Cusick’s conjecture:

Theorem 1 The Hausdorff dimension of ${(M\setminus L)\cap (3.7, 3.71)}$ is ${\geq 0.53128}$.

Below the fold, we give an outline of the proof of this theorem.

Remark 1 The basic reference for this post is the classical book of Cusick and Flahive.

1. Description of the key ideas

Recall that ${L=\{\limsup\limits_{n\rightarrow\infty} \lambda_n(a):a\in(\mathbb{N}^*)^{\mathbb{Z}}\}}$ and ${M=\{\sup\limits_{n\in\mathbb{Z}} \lambda_n(a):a\in(\mathbb{N}^*)^{\mathbb{Z}}\}}$, where

$\displaystyle \lambda_n((a_m)_{m\in\mathbb{Z}}) := [a_n; a_{n+1}, a_{n+2}, \dots] + [0;a_{n-1}, a_{n-2},\dots]$

and ${[b_0; b_1, \dots] = b_0+\frac{1}{b_1+\frac{1}{\ddots}}}$ is the usual continued fraction expansion.

We consider the finite word ${3322212}$ of odd length: this is a non semi-symmetric word in the sense of Flahive (i.e., it can not be decomposed into two palindromes).

Remark 2 The examples constructed by Freiman of elements in ${M\setminus L}$ were based on two non semi-symmetric words, and Flahive showed in her paper that “an element of ${M\setminus L}$ is often associated to non semi-symmetric words”.

The periodic sequence ${\alpha=\overline{3322212}}$ obtained by infinite concatenation of ${3322212}$ has Markov value ${\ell=\sup\limits_{n\in\mathbb{Z}} \lambda_n(\alpha) = [3;\overline{2221233}]+[0;\overline{3212223}] = 3.709699859679\dots}$

If we try to glue the word ${z=212121\dots}$ on the right of ${\alpha}$, we get a new sequence ${\gamma = \overline{3322212}33222123322212212121\dots}$ with Markov value

$\displaystyle \mu=\sup\limits_{n\in\mathbb{Z}} \lambda_n(\gamma)=3.70969985975\dots$

On the other hand, if we try to glue the word ${w=\dots212121}$ on the left of ${\alpha}$ in a way to obtain the smallest possible change in the Markov value, then the best choice is the sequence ${\delta = \dots212121221233222123322212\overline{3322212}}$ whose Markov value is

$\displaystyle \nu=\sup\limits_{n\in\mathbb{Z}} \lambda_n(\delta)=3.70969985982\dots$

Hence, the cost of gluing ${w}$ to the left of ${\alpha}$ is always higher than the cost of ${\gamma=\alpha z}$. This indicates that the Markov value ${\mu}$ of ${\gamma}$ doesn’t belong to ${L}$ because any attempt to reproduce ${\mu}$ as the ${\limsup}$ of ${\lambda_n(a)}$ for some sequence ${a\in\{1,2,3\}^{\mathbb{Z}}}$ would force the appearance of large chunks of ${\gamma = \alpha z}$ at arbitrarily large positions in ${a}$, so that ${a}$ would contain a subword (which is essentially) arbitrarily close to ${z\alpha}$ and, thus, its Markov value would be at least the Markov value ${\nu}$ of ${\delta}$, a contradiction (since ${\nu>\mu}$).

Closing this section, we remind for later use the following standard comparison lemma for continued fractions.

Lemma 2 Given ${\alpha=[a_0;a_1,\dots,a_n,a_{n+1},\dots]}$ and ${\beta=[a_0;a_1,\dots,a_n,b_{n+1},\dots]}$ with ${a_{n+1}\neq b_{n+1}}$, one has ${\alpha>\beta}$ if and only if ${(-1)^{n+1}(\alpha_{n+1}-\beta_{n+1})>0}$.

2. “Big” words and self-replication

The following lemma provides a list of words whose appearance in a sequence ${a\in\{1,2,3\}^{\mathbb{Z}}}$ forces its Markov value ${m(a):=\sup\limits_{n\in\mathbb{Z}} \lambda_n(a)}$ to be

$\displaystyle m(a)>3.70969985975033$

Lemma 3 If ${a=(a_n)_{n\in\mathbb{Z}} \in \{1,2,3\}^{\mathbb{Z}}}$ contain any of the words

• (1) ${3^*1}$
• (2) ${23^*2}$
• (3) ${33^*23}$
• (4) ${333^*22}$
• (5) ${233^*221}$
• (6) ${1233^*2223}$
• (7) ${21233^*2222}$
• (8) ${21233^*22211}$
• (9) ${111233^*2222}$
• (10) ${121233^*22212}$
• (11) ${3221233^*2221233}$
• (12) ${3211233^*222212}$
• (13) ${2221233^*22212333}$
• (14) ${2211233^*2222121}$ or ${2211233^*2222122}$
• (15) ${12221233^*22212332}$ or ${22221233^*22212332}$
• (16) ${12211233^*22221233}$ or ${22211233^*22221233}$
• (17) ${2332221233^*222123321}$
• (18) ${333211233^*22221112}$ or ${233211233^*22221112}$
• (19) ${22332221233^*222123322}$
• (20) ${12332221233^*2221233223}$
• (21) ${112332221233^*222123322212}$
• (22) ${3212332221233^*222123322212}$
• (23) ${12212332221233^*222123322212}$

and ${j\in\mathbb{Z}}$ is the position in asterisk, then

$\displaystyle \lambda_i(a)>3.70969985975033$

for some ${|i-j|\leq 17}$.

Proof: We prove this lemma by straightforward calculations using the standard comparison Lemma 2.

We verify the items in their order of appearance: for example, ${\lambda_j(\dots3^*1\dots)\geq [3;1,\overline{1,3}] + [0;\overline{3,1}] = 3.82\dots}$, ${\lambda_j(\dots23^*2\dots)\geq [3;2,\overline{1,3}] + [0;2,\overline{1,3}] = 3.71\dots}$, etc.

Sometimes, we need to use the previous items ((1), (2) and/or (5)) and the assumption that ${\lambda_i(a)\leq 3.70969985975033}$ for all ${1\leq |i-j|\leq 17}$ to get the desired conclusion for a given item: for instance, by (1) and the fact that ${\lambda_{j-1}(a)<3.70969985975033}$, we obtain (3):

$\displaystyle \lambda_j(\dots33^*23\dots)\geq [3;2,3,\overline{3,1}] + [0;3,2,\overline{3,1}] = 3.72\dots$

See our original article for more details or this Mathematica notebook here. $\Box$

A direct consequence of this lemma is the fact that the word

$\displaystyle 2332221233^*222123322$

must extend as

$\displaystyle 23322212332221233^*222123322212$

whenever the Markov value is ${\leq 3.70969985975033}$.

Corollary 4 Let ${a=\dots2332221233^*222123322\dots\in\{1,2,3\}^{\mathbb{Z}}}$ where ${j\in\mathbb{Z}}$ is the position in asterisk. If ${\lambda_i(a)\leq 3.70969985975033}$ for all ${|i-j|\leq 17}$, then ${a}$ extends as

$\displaystyle a=\dots23322212332221233^*222123322212\dots$

and the vicinity of the position ${j-7}$ is ${2332221233^*222123322}$.

Proof: By Lemma 3 (3) and (19), ${a}$ extends to the left as

$\displaystyle a=\dots12332221233^*222123322\dots$

By Lemma 3 (5) and (20), ${a}$ continues to the right as

$\displaystyle a=\dots12332221233^*2221233222\dots$

By Lemma 3 (6) and (7), ${a}$ extends to the right as

$\displaystyle a=\dots12332221233^*22212332221\dots$

By Lemma 3 (1) and (8), ${a}$ continues to the right as

$\displaystyle a=\dots12332221233^*222123322212\dots$

By Lemma 3 (1) and (21), ${a}$ extends to the left as

$\displaystyle a=\dots212332221233^*222123322212\dots$

By Lemma 3 (10) and (22), ${a}$ continues to the left as

$\displaystyle a=\dots2212332221233^*222123322212\dots$

By Lemma 3 (11) and (23), ${a}$ extends to the left as

$\displaystyle a=\dots22212332221233^*222123322212\dots$

By Lemma 3 (15), ${a}$ continues to the left as

$\displaystyle a=\dots322212332221233^*222123322212\dots$

By Lemma 3 (1), (2) and (4), ${a}$ extends to the left as

$\displaystyle a=\dots23322212332221233^*222123322212\dots$

This completes the proof. $\Box$

3. “Small” words and local uniqueness

The next lemma gives a list of words associated to positions where a Markov value ${> 3.70969985968}$ is not reached.

Lemma 5 If ${a=(a_n)_{n\in\mathbb{Z}} \in \{1,2,3\}^{\mathbb{Z}}}$ contain any of the words

• (24) ${33^*3}$
• (25) ${33^*21}$
• (26) ${233^*223}$
• (27) ${3233^*222}$ or ${2233^*222}$
• (28) ${11233^*2221}$
• (29) ${211233^*22223}$ or ${211233^*22222}$
• (30) ${321233^*22212}$
• (31) ${221233^*222121}$
• (32) ${1221233^*222122}$ or ${2221233^*222122}$
• (33) ${1221233^*2221233}$
• (34) ${1211233^*222211}$ or ${2211233^*222211}$
• (35) ${1211233^*222212}$
• (36) ${33211233^*2222112}$
• (37) ${2332211233^*222212333}$ or ${2332211233^*222212332}$
• (38) ${233211233^*22221111}$ or ${333211233^*22221111}$

and ${j\in\mathbb{Z}}$ is the position in asterisk, then

$\displaystyle \lambda_j(a)<3.70969985968$

Proof: The proof of this lemma is a straightforward calculation based on the standard comparison Lemma 2: see our original article or this Mathematica notebook here for more details. $\Box$

By putting together Lemmas 3 and 5, we show below that a Markov value between ${3.70969985968}$ and ${3.70969985975033}$ is necessarily attained at positions whose vicinities are

$\displaystyle 2332221233^*222123322$

Corollary 6 If ${a=(a_n)_{n\in\mathbb{Z}}\in \{1,2,3\}^{\mathbb{Z}}}$ satisfies

$\displaystyle 3.70969985968<\lambda_j(a)<3.70969985975033$

and ${\lambda_i(a)<3.70969985975033}$ for all ${|i-j|\leq 17}$, then, up to transposition about the ${j}$th position,

$\displaystyle a=\dots2332221233^*222123322\dots$

where the asterisk indicates the ${j}$th position.

Proof: By Lemma 3 (1) and (2), and Lemma 5 (24), we have that, up to transposition,

$\displaystyle a=\dots33^*2\dots$

By Lemma 3 (3) and Lemma 5 (25), ${a}$ extends to the right as

$\displaystyle a=\dots33^*22\dots$

By Lemma 3 (1) and (4), ${a}$ extends to the left as

$\displaystyle a=\dots233^*22\dots$

By Lemma 3 (5) and Lemma 5 (26), ${a}$ continues to the right as

$\displaystyle a=\dots233^*222\dots$

By Lemma 5 (27), ${a}$ is forced to extend to the left as

$\displaystyle a=\dots1233^*222\dots$

By Lemma 3 (1) and Lemma 5 (28), ${a}$ continues to the left as

$\displaystyle a=\dots21233^*222\dots$

By Lemma 3 (6), (7) and (8), ${a}$ is forced to continue on the right as

$\displaystyle a=\dots21233^*22212\dots$

By Lemma 3 (10) and Lemma 5 (30), ${a}$ extends on the left as

$\displaystyle a=\dots221233^*22212\dots$

By Lemma 5 (31) and Lemma 3 (1), (2), ${a}$ continues as

$\displaystyle a=\dots221233^*222122\dots$

or

$\displaystyle a=\dots221233^*2221233\dots$

We claim that the first possibility can’t occur. Indeed, by Lemma 5 (32) and Lemma 3 (1), (2) and (4), this word would extend to the left as

$\displaystyle \dots233221233^*222122\dots$

so that its left side would contain the word ${233221}$, a contradiction with Lemma 3 (5). Therefore, we have to analyse the word

$\displaystyle a=\dots221233^*2221233\dots$

By Lemma 3 (11) and Lemma 5 (33), ${a}$ extends on the left as

$\displaystyle a=\dots2221233^*2221233\dots$

By Lemma 3 (1) and (13), ${a}$ continues on the right as

$\displaystyle a=\dots2221233^*22212332\dots$

By Lemma 3 (15), ${a}$ is forced to extend on the left as

$\displaystyle a=\dots32221233^*22212332\dots$

By Lemma 3 (1), (2) and (4), ${a}$ continues on the left as

$\displaystyle a=\dots2332221233^*22212332\dots$

This ends the proof. $\Box$

4. Conclusions

The first consequence of our previous discussion is the absence of Lagrange spectrum in the interval with extremities ${3.70969985968}$ and ${3.70969985975033}$.

Proposition 7 ${L\cap (3.70969985968, 3.70969985975033) = \emptyset}$

Proof: Suppose that ${\ell\in L\cap (3.70969985968, 3.70969985975033)}$, say ${\ell = \limsup_{n\rightarrow+\infty}\lambda_n(a)}$ for some ${a\in\{1,2,3\}^{\mathbb{Z}}}$.

By definition, we can select ${N\in\mathbb{N}}$ and an increasing sequence ${n_k\rightarrow\infty}$ such that

$\displaystyle \lambda_i(a)<3.70969985975033$

for all ${i\geq N}$, and

$\displaystyle 3.70969985968<\lambda_{n_k}(a)<3.70969985975033$

for each ${k\in\mathbb{N}}$.

By applying Corollary 6 at the positions ${n_k}$ and Corollary 4 at adequate positions of the form ${n_k\pm 7m\geq N}$ with ${m\in\mathbb{N}}$, we deduce that

$\displaystyle \ell = \lambda_0(\overline{33^*22212})<3.70969985968$

a contradiction. (Here ${\overline{33^*22212}}$ denotes the periodic sequence obtained by infinite concatenation of the word ${33^*22212}$.) $\Box$

Hence, any Markov value in the interval with extremities ${3.70969985968}$ and ${3.70969985975033}$ does not belong to the Lagrange spectrum. As it turns out, it is not hard to construct a whole Cantor set of such values: in fact, it is not difficult to check that the Markov values of the sequences

$\displaystyle \overline{3322212}33^*2221233222122121212\theta, \quad \theta\in\{1,2\}^{\mathbb{N}}$

form a Cantor set ${C\subset M}$ isomorphic to ${C(2):=\{[0;\theta]:\theta\in\{1,2\}^{\mathbb{N}}\}}$ such that

$\displaystyle C\subset (3.70969985975024, 3.70969985975028)$

In particular, we deduce that

$\displaystyle (M\setminus L)\cap (3.7, 3.71)\supset C$

and the Hausdorff dimension of ${C}$ is equal to the Hausdorff dimension of ${C(2)}$. This completes the proof of Theorem 1 in view of the recent results of Jenkinson and Pollicott on the Hausdorff dimension of ${C(2)}$.

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