Weighted Simultaneous Rational Approximation

• The topic defines a framework that generalizes classical rational and Diophantine approximation through user-prescribed anisotropic weights controlling error decay.
• It integrates geometric, analytic, and algorithmic techniques to derive robust metric theorems and explicit formulas for weighted exponents in real and p-adic settings.
• Applications include multivariate approximation, least-squares rational optimization, and nonlinear manifold analyses using stabilized algorithms like VARPRO and AAA initialization.

Weighted simultaneous rational approximation generalizes classical rational and Diophantine approximation by introducing anisotropic, user-prescribed weights in the error criteria for approximating systems of numbers or functions. This framework spans areas ranging from parametric geometry of numbers, weighted Diophantine exponents, and metric approximation theorems to multivariate and functional least-squares rational approximants. The rigorous study of both metric and constructive aspects necessitates integrating geometric, analytic, and algorithmic perspectives, as evidenced by recent advances in both the real and $p$-adic settings.

1. Definitions and Foundational Problems

1.1 Weighted Simultaneous Rational Approximation

Given a collection of objects (real or $p$-adic numbers, vectors, or functions) $\boldsymbol{\xi} = (\xi_1, \dots, \xi_n)$ and a weight vector $w = (w_1, \dots, w_n)$ with $w_i > 0$ and often $\sum w_i = 1$, weighted simultaneous approximation quantifies the accuracy of rational or integer-coordinate approximants under the constraint that the permissible error in $\xi_i$ decays at a rate $X^{-w_i \mu}$, where $X$ is a height parameter and $\mu$ is the approximation exponent.

Typical forms include:

• Real setting: For $(\xi_1, \dots, \xi_n)$, require for infinitely many $q \in \mathbb{Z}_+$, existence of $(p_1, \dots, p_n) \in \mathbb{Z}^n$ such that $|q \xi_i - p_i| < q^{-w_i \mu}$ for each $i$.
• $p$-adic setting: For $\alpha \in \mathbb{Z}_p^n$, require the existence of $$(a_0, a_1, \dots, a_n)\in \mathbb{Z}_+ \times \mathbb{Z}^n$$ with $(a_0, a_i) = 1$ and $|\alpha_i - a_i / a_0|_p < \psi(a_0)^{w_i}$ for all $i$, for a chosen non-increasing function $\psi$ (Beresnevich et al., 2021).

1.2 Weighted Exponents and Size Functions

For inhomogeneous Diophantine approximation (e.g., to $(1, \xi, \xi^2)$), the weighted norm of an integer vector $x = (x_0, x_1, x_2)$ is defined by $\|x\|_w = \max\{|x_1|^{1/w_1}, |x_2|^{1/w_2}\}$, with weighted exponents

$$\omega_w(\xi) = \sup\left\{\mu > 0 : \exists^\infty\,x\in\mathbb{Z}^3\setminus\{0\} \text{ with } |x_0 + x_1 \xi + x_2 \xi^2| \leq \|x\|_w^{-\mu}\right\},$$

and similarly for the uniform exponent $\widehat\omega_w(\xi)$ (Roy, 1 Feb 2026).

2. Metric and Geometric Results in Weighted Simultaneous Approximation

Weighted simultaneous rational approximation in both real and $p$-adic fields admits a comprehensive metric theory paralleling classical results.

2.1 Metric Theorems in the $p$-adic Case

In $p$-adic fields, the Lebesgue measure analog is Haar measure $U_p$ on $\mathbb{Z}_p^n$. Results include:

• Weighted $p$-adic Khintchine Theorem: For weights $w$ and error functions $\psi_i(q) = \psi(q)^{w_i}$, the set $W_n(\psi, w)$ of $(\psi, w)$-approximable points has full or zero measure according as $\sum_q \Psi(q)$ diverges or converges, with $\Psi(q) = \prod_{i=1}^n \psi_i(q)$ (Beresnevich et al., 2021).
• Weighted $p$-adic Duffin–Schaeffer Theorem: A zero-one law for $\limsup$ sets under mild density hypotheses, critical for nonmonotone $\psi$.
• Jarník–Besicovitch Theorem (Hausdorff dimension): For $T_i > 1$, $\sum T_i > n+1$,

$$\dim\,W_n(T) = \min_{1\leq k \leq n} \frac{n+1 + \sum_{i=1}^n \max\{0, T_k - T_i\}}{T_k}.$$

In the equal-exponent case, this becomes $(n+1)/T$ (Beresnevich et al., 2021).

The proofs utilize the Mass Transference Principle adapted to $p$-adic geometry, including "Rectangles→Rectangles" versions to handle anisotropic scaling.

2.2 Weighted Jarník Identity and Inequalities in the Real Case

For simultaneous approximation of $(\xi_1, \xi_2)$ with weights $\nu_1, \nu_2 > 0$, $\nu_1 + \nu_2 = 1$, the weighted exponents $\omega_s, \hat{\omega}_s$ (simultaneous) and $\omega_\ell, \hat{\omega}_\ell$ (linear forms) satisfy German's inequalities: $$(1+\nu_2)\hat{\omega}_s + (1+\nu_1)\omega_\ell \leq \hat{\omega}_s\omega_\ell + 2 \nu_1 \nu_2, \ (1+\nu_1)\hat{\omega}_s + (1+\nu_2)\omega_\ell \geq \hat{\omega}_s\omega_\ell + 2 \nu_1 \nu_2$$ with classical Jarník identity as a special case for $\nu_1 = \nu_2 = 1/2$ (Summerer, 2019).

2.3 Multi-parametric Geometry of Numbers and Successive Minima

The multi-parametric (weighted) geometry-of-numbers framework uses one-parameter families of convex bodies with facet expansion/shrinkage dictated by the weights. Successive minima functions $L_j(q)$, the logarithms of the successive minima of these convex bodies, encode the Diophantine exponents via

$$\frac{1}{1+\omega_w(\xi)} = \liminf_{q\to\infty} \frac{L_1(q)}{q},\quad \frac{1}{1+\widehat\omega_w(\xi)} = \limsup_{q\to\infty} \frac{L_1(q)}{q}$$

(Roy, 1 Feb 2026, Summerer, 2019). For extremal real numbers, this structure allows closed-form formulas for the weighted exponents.

3. Explicit Results: Extremal Numbers and Weighted Exponents

Damien Roy established an explicit computation of the exponents for extremal real numbers $\xi$, i.e., numbers for which the system $(1, \xi, \xi^2)$ exhibits periodic behavior in the associated geometry-of-numbers framework. For weights $w = (w_1, w_2)$,

$$\omega_w(\xi) = \frac{w_2 + \Theta w_1}{w_1 + \Theta w_2},\quad \widehat\omega_w(\xi) = \frac{w_2 + \theta w_1}{w_1 + \theta w_2}$$

where $\Theta = (\sqrt{5} + 1)/2$ and $\theta = (\sqrt{5} - 1)/2$ (Roy, 1 Feb 2026). These furnish a continuous interpolation between classical single-variable exponent extremality and full simultaneous approximation.

When $w = (1,0)$, the result specializes to $\widehat\omega_{(1,0)}(\xi) = \theta$, $\omega_{(1,0)}(\xi) = \Theta$, recovering the ordinary exponents for extremal numbers.

4. Weighted Least Squares Rational Approximation and Algorithms

Weighted simultaneous rational approximation in function spaces (e.g., of transfer functions) leads to nonlinear optimization problems with a weighted $\ell_2$-norm objective. Formally, for given data $f(z_j)$ and weighting matrix $W$, one seeks rational functions $r(z)$ (in polynomial-ratio or partial-fraction form) minimizing

$\min_{r} \| W^{1/2}(f - r(Z)) \|_2^2.$

(Hokanson et al., 2018)

Parameterizations and Algorithmic Methods

• Polynomial-Ratio Parameterization: $r(z) = n(z)/d(z)$ with $n(z), d(z)$ expanded in appropriate polynomial bases. Nonlinear least squares is solved in both numerator and denominator coefficients.
• Partial Fraction (Pole-Residue) Parameterization: $r(z) = \sum_k \rho_k/(z - \lambda_k) +$ polynomial tail. The optimization proceeds over pole locations, residues, and tail coefficients.
• Variable Projection (VARPRO): This methodology eliminates linear parameters (e.g., residues, polynomial coefficients) at each step, reducing the problem to nonlinear optimization in a lower-dimensional space (e.g., poles, denominator roots).

Algorithmically, Gauss–Newton with Levenberg–Marquardt damping is employed, while initialization via the AAA (Adaptive Antoulas–Anderson) algorithm provides reliable starting points, especially important for non-convex landscapes. For real-rational approximation, parameterizations enforcing real coefficients or pairing complex-conjugate poles stabilize the optimization (Hokanson et al., 2018).

Stability and Conditioning

Empirical studies demonstrate that the partial-fraction basis yields substantially better numerical conditioning than polynomial-ratio forms, and AAA-based initialization plus careful factorization of $W$ mitigates local minima and ill-conditioning (Hokanson et al., 2018).

5. Multivariate and High-Dimensional Aspects

The weighted approximation paradigm extends to multivariate contexts (either multiple real/p-adic variables or higher-dimensional function fitting). For rational approximation problems in higher dimensions or multiple variables, the stabilized Sanathanan-Koerner (S-SK) iteration, using Arnoldi-based Vandermonde constructions, addresses severe numerical instability inherent in the classical SK iteration for high degrees or clustering (Hokanson, 2020).

Algorithmically, for each iteration:

• Construct orthonormal polynomial bases via Arnoldi iteration adapted to the current weights.
• Solve block least-squares problems in these orthonormal bases, which are robust to ill-conditioning.
• Update weights and iterate, with convergence typically observed to be linear.

In parametric model reduction and multivariate approximation, such stabilized algorithms yield smaller residuals and more uniform accuracy than previous methods, for both unweighted and weighted cases (Hokanson, 2020).

6. Applications to Manifolds and Metric Theory

Weighted simultaneous approximation extends beyond vector settings to nonlinear embedded manifolds. In the $p$-adic context, for $C^2$ or DQE maps $f: U \rightarrow \mathbb{Z}_p^m$, metric theorems ensure:

• Dirichlet-type existence: Almost every $x$ satisfies infinitely many rational approximations at decay rates determined by prescribed weights.
• Mass Transference Principle: Used to transfer full-measure statements into Hausdorff dimension lower bounds for sets of well-approximable points on submanifolds.
• Dimension Formulas: For $F: U\to \mathbb{Z}_p^n$ an embedding, the dimension of the inverse image of weighted well-approximable sets $F^{-1}(W_n(T))$ is

$$\dim F^{-1}(W_n(T)) \geq \min_{1\leq k\leq d} \frac{n+1 + \sum_{i=1}^n \max\{0, T_k - T_i\}-m}{T_k},$$

which parallels the classical Jarník–Besicovitch result but in the weighted $p$-adic manifold context (Beresnevich et al., 2021).

7. Technical Constraints, Uniqueness, and Open Directions

Distinct features arise in the weighted, simultaneous, and $p$-adic settings:

• Anisotropy and Weight Constraints: Constraints such as $T_i>1$ or $\psi_i(q)<1$ arise directly from the structure of $\mathbb{Q}_p$, as points can be arbitrarily close $p$-adically.
• Unique Zero-One Laws: The ultrametric geometry requires specialized covering arguments and density lemmas for establishing zero-one laws in limsup measure settings.
• Extensions: Recent work strongly suggests that the multi-parametric geometry-of-numbers and regular slope-system techniques will allow higher-dimensional weighted analogues of Jarník-type identities and explicit exponent computations for a broader class of vectors and manifolds (Roy, 1 Feb 2026, Summerer, 2019).
• Algorithmic Limitations: Fixed-point and iterative methods (e.g., SK iteration) can be severely ill-conditioned for high degrees (e.g., $n>15$), supporting the need for partial fraction and stabilized algorithms (Hokanson et al., 2018, Hokanson, 2020).

Weighted simultaneous rational approximation reveals rich structures—both metric and constructive—that unify geometric, analytic, and computational themes across classical real, $p$-adic, and algorithmic settings. The interplay of weights introduces a nuanced control over approximation phenomena and conditions, and recent advances yield both effective computational algorithms and explicit Diophantine bounds, with ongoing developments anticipated for high-dimensional and dynamical generalizations (Summerer, 2019, Beresnevich et al., 2021, Roy, 1 Feb 2026, Hokanson et al., 2018, Hokanson, 2020).