Freiman's constant

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Number

$c_F$

4.527829566160879140882695988070469646929833632769728374065061792009862983413135427323086150717070952

Definition

Freiman's constant $c_F$ is the endpoint of the last gap in the Lagrange spectrum [3]. It is the quadratic irrational in Formula (1).

Formulas

(1)

$c_F=\dfrac{2221564096+283748\sqrt{462}}{491993569}$.

(2)

$491993569c_F^2-4443128192c_F+10031248672=0$, where $c_F$ is the larger root.

Comments

(3)

The half-line $[c_F,\infty)$ is often called Hall's ray. Every real number at least $c_F$ lies in both the Markov and Lagrange spectra [1] [4]; this table stores the endpoint, not the continuum of all values on the ray.

(4)

The discrete part of the Markov and Lagrange spectra with value less than $3$ is stored in Lagrange numbers $L_m$ of the Markov spectrum.

(5)

Freiman proved the endpoint result in his monograph [2]; Cusick and Flahive give the value and spectrum context [1].

Programs

(P1)

SageMath

R = RealBallField(400)
(R(2221564096) + R(283748) * R(462).sqrt()) / R(491993569)

References

[1]

T. W. Cusick and M. E. Flahive, The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs 30, American Mathematical Society, 1989.

[2]

G. A. Freiman, Diophantine approximations and the geometry of numbers (Markov's problem), Kalinin State University Press, Kalinin, 1975.

Links

[3]

MathWorld: Freiman's Constant

[4]

Wikipedia: Markov spectrum

[5]

OEIS A118472: decimal expansion of Freiman's constant

Similar tables

Lagrange numbers $L_m$ of the Markov spectrum — store the discrete Markov and Lagrange spectrum values less than $3$

Khinchin's means — describe almost-sure limits of regular continued fractions, while Freiman's constant is an extremal threshold

Lévy's constant — describes the almost-sure denominator growth rate for regular continued fractions, while Freiman's constant is an extremal threshold for the Lagrange spectrum

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: yes

How they were obtained:

The generator computes Formula (1) in real ball arithmetic from exact integers. The stored decimal was compared with OEIS A118472 [5], and the computed ball was checked against the minimal polynomial in Formula (2).