---
title: Non-Hitting Spectrum in Markov–Lagrange Setting
url: https://www.emergentmind.com/topics/non-hitting-spectrum
type: topic
---

# Non-Hitting Spectrum in Markov–Lagrange Setting

Searching arXiv for the primary and closely related papers on the non-hitting spectrum in the Markov–Lagrange setting.
In Diophantine approximation, the **non-hitting spectrum** denotes the set
\[
M\setminus L,
\]
the complement of the Lagrange spectrum \(L\) inside the Markov spectrum \(M\). Equivalently, these are Markov values that do not arise as a \(\limsup\) in Perron’s continued-fraction description of \(L\). Although \(M\setminus L\) was long known primarily through isolated examples, work of Freiman, Cusick–Flahive, and later Matheus–Moreira established that it contains structured fractal pieces of positive Hausdorff dimension, including a Cantor model near a non-isolated point \(\alpha_\infty\), together with explicit new elements and quantitative lower bounds such as \(HD(M\setminus L)>0.353\) [1703.04302]. Complementary analysis around Freiman’s isolated points identified the largest interval \((c_\infty,C_\infty)\) containing Freiman’s countable family and avoiding \(L\), located the smallest known element \(f\) in that interval, and proved a separate positive dimension bound \(>0.2628\) for the corresponding portion of the Markov spectrum [1802.02454].

## 1. Classical definitions and Perron’s description

For an irrational \(x\), the classical Diophantine approximation constant is
\[
L(x) \;=\;\limsup_{q\to\infty}\;q\;\|q\,x\|\,,
\]
where \(\|\,\cdot\,\|\) is the distance to the nearest integer. The Lagrange spectrum is
\[
L\;=\;\{\,L(x)\;>\;0:\;x\in\mathbb R\setminus\mathbb Q\}\,.
\]

The Markov spectrum admits an equivalent formulation in terms of indefinite binary quadratic forms of determinant \(1\):
\[
M\;=\;\Bigl\{\;\frac1{\inf_{(m,n)\in\mathbb Z^2\setminus\{0\}|ax^2+bxy+cy^2|}\;>\;0:
a,b,c\in\mathbb R,\;b^2-4ac=1\Bigr\}\,.
\]

A convenient description is Perron’s continued-fraction characterization. For a bi-infinite sequence
\[
A=(\dots,a_{-1},a_0,a_1,\dots)\in(\mathbb N^*)^\mathbb Z,
\]
define
\[
X_i(A)\;=\;\bigl[\,a_i;\,a_{i+1},a_{i+2},\dots\big]  \;+\;\bigl[\,0;\,a_{i-1},a_{i-2},\dots\big]
\quad(i\in\mathbb Z).
\]
Then
\[
L\;=\;\{\limsup_{i\to\infty}X_i(A)\},\qquad
M\;=\;\{\sup_{i\in\mathbb Z}X_i(A)\}.
\]
It follows that \(L\subset M\), and both are closed subsets of \([\sqrt5,\infty)\) [1703.04302]. In the terminology used in the later survey of Freiman’s isolated points, the set
\[
M\setminus L \;=\; \{\,m\in M: m\notin L\}
\]
consists exactly of Markov values that “never hit” the Lagrange spectrum, which motivates the name **non-hitting spectrum** [1802.02454].

The distinction between \(\sup\) and \(\limsup\) is decisive. A Markov value may occur as the global maximum of the quantities \(X_i(A)\) along a bi-infinite coding without recurring in the asymptotic sense required for membership in \(L\). This suggests a symbolic-dynamical viewpoint in which \(M\setminus L\) records exceptional maxima constrained by finite combinatorics of continued-fraction words.

## 2. Freiman’s breakthrough and the first non-isolated point

Freiman proved in 1973 that \(M\setminus L\neq\varnothing\) by exhibiting
\[
\alpha_\infty = [2;1,2,2,3,1,2]  +  [0;1,2,3,1,2,1,2] \approx3.2930442\ldots
\]
with \(\alpha_\infty\in M\setminus L\) [1703.04302]. Cusick–Flahive later produced an infinite sequence
\[
a_n\to\alpha_\infty \qquad (n\to\infty),
\]
all lying in \(M\setminus L\), so \(\alpha_\infty\) is a non-isolated point of the non-hitting spectrum [1703.04302].

A different and earlier strand of Freiman’s work produced isolated points of \(M\setminus L\). In 1968 he constructed an explicit bi-infinite sequence \(F\) with Markov value
\[
\mathfrak o \;=\; m(F) \;=\; \bigl[2;4,1,2,2,1\bigr] \;+\; \bigl[0;1,2,2,2,1,2,4,1,2,2,1,2,2,1,2,2,1\bigr] \;\approx\;3.1181,
\]
and generalized this to an infinite countable set
\[
\mathcal F=\{\,m(F_w)\:w\in\{\text{finite words in }1,2\}\} \subset M\setminus L
\]
whose elements are isolated in \(M\) [1802.02454].

These two phenomena—isolated points and a non-isolated accumulation point—already show that \(M\setminus L\) is not a uniform object. One part is discrete and countable; another supports accumulation and, as later work proved, positive Hausdorff dimension. A common misconception is that \(M\setminus L\) is merely a sporadic exceptional set. The existence of \(\alpha_\infty\) and its surrounding Cantor structure rules this out [1703.04302].

## 3. Cantor model near \(\alpha_\infty\)

A refinement of Perron-type arguments shows that any \(m\in M\setminus L\) sufficiently close to \(\alpha_\infty\) arises from a bi-infinite continued-fraction sequence on the alphabet \(\{1,2\}\) that avoids a finite forbidden list
\[
P =\{\,21212,\,21213,\,13212,\,121212,\,122121,\,231212221,\,
122122123,\,123121222,\,221221231\}\,.
\]
This leads to the dynamically defined Cantor set
\[
X
=\Bigl\{x=[0;a_1,a_2,\dots]:\;a_i\in\{1,2\},\;\text{no subblock from }P\Bigr\}.
\]

Matheus–Moreira proved that
\[
(M\setminus L)\cap(b_\infty,B_\infty)
=\text{a countable union of sets all diffeomorphic to }X,
\]
where \((b_\infty,B_\infty)\) is the maximal open interval around \(\alpha_\infty\) disjoint from \(L\). In particular,
\[
\dim_H\bigl(M\setminus L\cap(b_\infty,B_\infty)\bigr)=\dim_H(X)
\]
[1703.04302].

This result turns a local problem in the Markov spectrum into a finite-type symbolic system. The crucial feature is that the non-hitting condition near \(\alpha_\infty\) can be encoded by forbidding finitely many words, so the local geometry becomes that of a Gauss-type Cantor set. This suggests that at least some parts of \(M\setminus L\) are naturally modeled by subshifts of finite type under continued-fraction coding.

The same structural principle reappears in the later study of Freiman’s isolated region. There, every Markov value in \((c_\infty,C_\infty)\) arises from a continued fraction in \(\{1,2\}^{\mathbb N}\) avoiding a finite collection of forbidden words, and the corresponding Cantor set
\[
Y =\bigl\{x=[0;a_1,a_2,\dots]\in[0,1]\;:\;a_i\in\{1,2\},\; \text{\(x\) contains none of the 27 forbidden blocks}\bigr\}
\]
satisfies
\[
\dim_H\bigl(M\cap(c_\infty,C_\infty)\bigr) = \dim_H(Y)
\]
[1802.02454]. The two constructions differ in local combinatorics but share the same mechanism: forbidden subwords produce a Gauss-Cantor model for a fragment of the non-hitting spectrum.

## 4. Hausdorff dimension and thermodynamical formalism

The Cantor set \(X\) is a dynamically defined Cantor set for the Gauss map \(G(x)=\{1/x\}\mod 1\) with inverse branches
\[
\phi_1(x)=\frac1{1+x},\qquad
\phi_2(x)=\frac1{2+x}.
\]
For words of length \(n\) avoiding \(P\), let
\[
\mathcal R_n=\{\phi_{i_1}\circ\phi_{i_2}\circ\cdots\circ\phi_{i_n}(I):\;i_j\in\{1,2\},\text{ no }P\text{-subblock}\},
\]
and for each \(R\in\mathcal R_n\) define
\[
m_{n,R}=\inf_{x\in R}|(\phi_{i_1\cdots i_n})'(x)|,\qquad
M_{n,R}=\sup_{x\in R}|(\phi_{i_1\cdots i_n})'(x)|.
\]
Then set
\[
a_n=\min_{R\in R_n}m_{n,R},\qquad
b_n=\max_{R\in R_n}M_{n,R}.
\]
Using Palis–Takens estimates, for every \(n\),
\[
\frac{\log|R_n|}{-\log a_n}\;\le\;\dim_H(X)\;\le\;\frac{\log|R_n|}{-\log b_n}.
\]
A computer-assisted evaluation at \(n=12\) yields
\[
0.353\;<\;\dim_H(X)\;<\;0.35792
\]
[1703.04302].

A simpler rigorous lower bound comes from the inclusion
\[
X\supset K(\{1,22\}),
\]
where \(K(\{1,22\})\) is the Gauss-Cantor set generated by \(\phi_1\) and \(\phi_{22}=\phi_2\circ\phi_2\). For the 12th iterate one finds
\[
a_{12}( \{1,22\} )\approx0.353465,\qquad
b_{12}( \{1,22\} )\approx0.357917,
\]
hence
\[
0.353<\dim_H\bigl(K(\{1,22\})\bigr)\le\dim_H(X).
\]
Therefore
\[
HD(M\setminus L)>0.353
\]
[1703.04302].

A parallel dimension argument applies in the interval \((c_\infty,C_\infty)\). There the relevant Cantor set \(Y\) contains
\[
K(\{12,22\}),
\]
and a computer-assisted pressure computation gives
\[
0.2628\;<\;\dim_H\bigl(K(\{12,22\})\bigr) \;\le\; 0.2646.
\]
Since \(K(\{12,22\})\subset Y\subset M\cap(c_\infty,C_\infty)\), one concludes
\[
\dim_H\bigl(M\cap(c_\infty,C_\infty)\bigr) \;>\;0.2628
\]
[1802.02454].

These results establish positive local thickness without claiming that the entire set \(M\setminus L\) has been completely characterized. The available bounds are local and constructive. A plausible implication is that finite-type symbolic restrictions provide a systematic route to quantifying Hausdorff dimension in selected windows of \(M\setminus L\).

## 5. Explicit elements, extremal known points, and local intervals

Matheus–Moreira’s symbolic extensions produce infinitely many new values in \(M\setminus L\) accumulating at \(b_\infty\). In particular, they give the largest known member
\[
c =\;[\,2;12,23,1,2\,] \;+\;[\,0;1,23,12,2,1,22,1,23,1,2,12,2,12,2\,]\;\approx3.29304447990138
\]
which exceeds the former record \(a_4=3.29304427\ldots\) of Cusick–Flahive [1703.04302].

On the side of Freiman’s isolated region, there is a maximal open interval
\[
(c_\infty,C_\infty)\subset\mathbb R
\]
containing \(\mathcal F\) and disjoint from \(L\), with endpoints
\[
c_\infty= \bigl[\,2;4,1,2,2,1\bigr] \;+\; \bigl[\,0;1,2,2,2,1,2,4\,\bigr]
 = 3.11812017814369\ldots,
\]
\[
C_\infty= \bigl[\,2;1,2,2,1,2,4,1,2,2,1,2,2,1,2,4,1,3\bigr]
 \;+\; \bigl[\,0;2,3,1,2,2,1,2,4,1,3\bigr]
 = 3.118120178328746016\ldots
\]
[1802.02454].

Within this interval, the smallest known element of \(M\setminus L\) is
\[
f \;=\; m(p) \;=\; \bigl[2;4,1,2,2,1\bigr] \;+\; \bigl[0;1,2,2,2,1,2,2,1,2,2,3,1,3\bigr]
 \;=\; 3.11812017815984\ldots
\]
and
\[
f=\min\bigl\{M\setminus L\cap(c_\infty,C_\infty)\bigr\}
\]
[1802.02454].

These computations show that the non-hitting spectrum is accessible not only via abstract dimension theory but also through explicit continued-fraction formulas. In this sense, the subject combines combinatorial symbolic dynamics with concrete arithmetic data: one can identify forbidden patterns, compute fractal dimensions, and still exhibit named extremal elements by exact expansions.

## 6. Geometric picture and significance

Several geometric conclusions emerge from these works. First, \(M\setminus L\) is not exhausted by a countable family of isolated exceptions. Near \(\alpha_\infty\), it is homeomorphic via continued-fraction coding to a Cantor set \(X\) of positive Hausdorff dimension [1703.04302]. Second, around Freiman’s isolated points, the Markov spectrum inside \((c_\infty,C_\infty)\) also carries a Cantor-type structure arising from forbidden words, with positive dimension \(>0.2628\) [1802.02454].

The overarching mechanism is uniform across these analyses. One starts from Perron’s formulas, imposes local constraints guaranteeing exclusion from \(L\), translates these constraints into finitely many forbidden subwords on the alphabet \(\{1,2\}\), and then studies the resulting Gauss-Cantor set by thermodynamical or pressure-type estimates. This gives a symbolic-dynamical explanation of why the non-hitting spectrum can be fractal rather than discrete.

A common misunderstanding is to identify the interval disjoint from \(L\) with a gap in \(M\). The cited results do not say that the relevant intervals are empty of Markov values. On the contrary, \((b_\infty,B_\infty)\) and \((c_\infty,C_\infty)\) are intervals avoiding \(L\) but supporting intricate subsets of \(M\), including Cantor pieces and explicit elements of \(M\setminus L\) [1703.04302; 1802.02454].

From a broader perspective, the non-hitting spectrum sits at the intersection of Diophantine approximation, continued fractions, hyperbolic dynamics, and fractal geometry. Its study converts arithmetic questions about best approximation constants into dimension-theoretic questions about symbolic subshifts and dynamically defined Cantor sets. This suggests that further progress is likely to depend on increasingly refined symbolic codings and rigorous computer-assisted estimates for the associated pressure equations.

Source: https://www.emergentmind.com/topics/non-hitting-spectrum