Extremal Real Numbers in Diophantine Approximation

• Extremal real numbers are transcendental numbers with provably optimal Diophantine approximation properties, defining strict bounds for simultaneous approximation of (ξ, ξ²).
• Their characterization employs matrix recurrences and the parametric geometry of numbers to precisely determine critical exponents in low dimensions.
• They serve as pivotal examples in the Markoff spectrum, illustrating rigidity phenomena and optimality thresholds in polynomial and rational approximations.

An extremal real number is a transcendental number possessing provably optimal Diophantine approximation properties—specifically, it meets certain bounds for simultaneous rational (and polynomial) approximation that cannot be improved, and it often serves as a critical case for deep theorems in the geometry of numbers. This concept arises in the context of understanding the best possible exponents achievable in simultaneous approximation to powers of a real number by rationals or algebraic numbers, with connections to the Markoff spectrum, parametric geometry of numbers, and the fine structure of approximation constants. Results by Roy, Davenport–Schmidt, Bugeaud–Laurent, Schleischitz, and others precisely characterize extremal numbers and delineate their exceptional properties, especially in low dimensions.

1. Formal Definition and Core Characterization

A real number $\xi\notin\mathbb{Q}$ is called extremal (in the sense of Davenport–Schmidt) if it is not quadratic over $\mathbb{Q}$ and the optimal exponent $1/\gamma$ (where $\gamma = (1+\sqrt{5})/2$ is the golden ratio) for simultaneous approximation by rationals to $(\xi, \xi^2)$ cannot be improved. Specifically, there exists $c>0$ such that, for all large $X$, there is a nonzero $(x_0, x_1, x_2)\in\mathbb{Z}^3$ with

$$|x_0| \leq X, \quad |x_0 \xi - x_1| \leq c X^{-1/\gamma}, \quad |x_0 \xi^2 - x_2| \leq c X^{-1/\gamma}.$$

Any attempt to increase the exponent $1/\gamma$ fails for extremal $\xi$ (Roy et al., 2010). Roy's definition, in a slightly different form, postulates that $\zeta$ is extremal if the uniform simultaneous exponent $\hat\lambda_2(\zeta) = (3+\sqrt{5})/2$ (Schleischitz, 2016).

A Markoff extremal number is an extremal number $\xi$ for which the Lagrange constant attains its maximum among non-quadratic irrationals: $$v(\xi) = \liminf_{n\to\infty} n\lVert n\xi\rVert = 1/3$$.

2. Approximation Exponents and Optimality

The classical irrationality measure is generalized in higher dimensions by the exponents $\lambda_n(\zeta)$, $\hat\lambda_n(\zeta)$ (rational approximation exponents), and the dual polynomial exponents $w_n(\zeta)$, $\hat{w}_n(\zeta)$, which measure the quality of approximation to $(\zeta, \zeta^2, ..., \zeta^n)$ by rationals or to $\zeta$ by algebraic numbers of fixed degree.

For extremal $\zeta$:

• In dimension $n=2$,

$$\lambda_2(\zeta) = \gamma, \quad \hat\lambda_2(\zeta) = (3+\sqrt{5})/2,$$

$$w_2(\zeta) = 2+\sqrt{5}, \quad \hat{w}_2(\zeta) = \gamma,$$

with higher successive minima precisely determined (Schleischitz, 2016).

• For $n=3$, Schleischitz determines that for extremal $\zeta$,

$$w_3(\zeta) = \hat{w}_3(\zeta) = 3, \quad \lambda_3(\zeta) = 1/(2+\sqrt{5}), \quad \hat\lambda_3(\zeta) = 1/3.$$

These exponents—critical in Diophantine approximation—show that extremal numbers achieve the theoretical limit for the rate at which $(1, \xi, \xi^2)$ (or higher powers) can be approximated by rationals or low-degree algebraic numbers.

For algebraic approximation to Markoff extremal numbers, the inequalities

$|\theta - P(\xi)| \le c|P|^{-\gamma}$

(DS–BL bound) are optimal: no better exponent than $\gamma$ is possible for $\theta = R(\xi)$ with $R$ of degree $d \in \{3,4,5\}$. The corresponding bound for roots $\alpha$ of $P+R$ (algebraic of degree $d$) is $|\xi - \alpha| \geq c_2 H(\alpha)^{-\gamma-1}$, which is also essentially optimal (Roy et al., 2010).

3. Matrix Recurrences, Geometry of Numbers, and Spectral Structure

Markoff extremal numbers can be characterized algebraically via sequences of symmetric matrices $X_k \in SL_2(\mathbb{Z})$, generated by a three-term recurrence,

$$X_{k+2} = X_{k+1} M_{k+1} X_k, \quad M_k = (-1)^{k+1}\begin{pmatrix}3 & 1\ -1 & 0\end{pmatrix}, \quad |X_{k+1}| \asymp |X_k|^\gamma,$$

with each $X_k$ connected to the simultaneous approximations above (Roy et al., 2010).

The parametric geometry of numbers (Schmidt–Summerer theory) frames these properties in terms of successive minima functions and their piecewise-linear graphs, as detailed for $(\zeta, \zeta^2, \zeta^3)$, capturing the alternation of optimal slopes and measuring the deviation from Dirichlet's principle (Schleischitz, 2016).

4. Dimension-Specific Phenomena and Breakdown of Extremality

While exponents for extremal numbers are optimal—and in fact dictate the best possible bounds—for approximations by polynomials of degree $d=3,4,5$, the phenomenon shifts at $d=6$. For Markoff extremal $\xi$, there exist infinitely many algebraic numbers of degree $6$ approximating $\xi$ with a bound that slightly beats the exponent $\gamma+1$, namely,

$$|\xi-\alpha| < C H(\alpha)^{-\gamma-1}(\log\log H(\alpha))^{-1},$$

for special forms of degree-$6$ polynomials (Roy et al., 2010). This breakdown indicates nuanced Diophantine phenomena specific to degree $6$ and possibly higher.

An outline of the proof strategy involves:

• Exploiting recurrence relations among $X_k$ to analyze fractional parts $\{x_{k,0}R(\xi)\}$, showing periodicity and separation from zero, thus establishing lower bounds.
• For $d=6$, explicit construction of nearly vanishing polynomials $P_k(T)$ leveraging this periodicity, leading to the improved approximation rate.

5. Role in the Markoff Spectrum and Diophantine Extremality

Markoff extremal numbers form a countable subset within the transcendental part of the Markoff spectrum (those $\xi$ with $v(\xi)=1/3$), representing transcendental numbers with "most quadratic-like" irrationality among all non-quadratic reals (Roy et al., 2010). They occupy a privileged position in the fine structure of Diophantine approximation, dictating the sharp thresholds for approximation constants in various dimensions and polynomial degrees.

These numbers also exhibit rigidity: their exponents align with the boundaries of Khintchine’s transference inequalities, and achieving equality in these inequalities (for any $n$) enforces that the corresponding uniform exponents reach their extremal values (Schleischitz, 2016).

6. Constructed Examples and Unresolved Directions

Explicit closed-form examples of extremal numbers are not known; constructions invoke continued fraction expansions with partial quotients growing according to a Fibonacci-type recursion, leading to numbers with the required matrix recurrence structure (Schleischitz, 2016). Computational and analytic exploration of these examples supports the observed periodic block structure in the parametric geometry of numbers.

Key open problems include establishing the full spectrum of exponents for $n \geq 4$, verifying conjectures such as $w_4(\zeta) = 2+\sqrt{5}$ for all extremal $\zeta$, and extending constructions and techniques—particularly combining recurrences with parametric geometry of numbers—to higher dimensions.

7. Broader Implications and Connections

Extremal real numbers provide both extremal counterexamples and sharp benchmarks for problems in Diophantine approximation theory, including:

• Optimality (up to constants) of approximation bounds for algebraic numbers of degrees $3,4,5$.
• Structural insight into the Markoff/Diophantine spectra.
• Connections to weighted simultaneous approximation, parametric geometry of numbers, and the algebraic theory of transcendence.

Their study illuminates rigidity phenomena, exemplifies the boundary cases for fundamental inequalities, and guides generalizations in higher-dimensional and multi-parameter settings (Roy et al., 2010, Schleischitz, 2016).