Diophantine Properties of Lie Groups

• Diophantine properties of Lie groups are defined by criteria that quantify how closely group elements can be approximated by discrete (integer or rational) subsets.
• The theory connects metric Diophantine approximation, Lie group structure, and representation theory to establish explicit approximation rates and identify algebraic obstructions.
• Analyses span nilpotent, solvable, and reductive groups, leveraging explicit exponents, rigidity results, and decision problems to advance applications in dynamics and differential operators.

Diophantine properties of Lie groups refer to deep quantitative phenomena governing the approximation of group elements (or orbits) by discrete subsets, typically integer or rational points, and the algebraic and geometric obstructions which control such approximation rates. The theory connects number-theoretic metric Diophantine approximation, the algebraic structure of Lie groups, representation theory, and dynamics on homogeneous spaces. The core objects of study are Diophantine subgroups, critical exponents, algebraic barriers to extremality, and the relationship between Diophantine decision problems in groups and their base rings.

1. Metric Diophantine Approximation and Group Structure

Given a real Lie group $G$ equipped with a left-invariant Riemannian metric, the Diophantine properties are usually formulated with respect to finitely generated subgroups $\Gamma \leq G$ and their generating sets $S$. For $n \in \mathbb{N}$, let $B_\Gamma(n)$ denote the word ball of radius $n$. $\Gamma$ is called $\beta$-Diophantine if there exists $c>0$ so that, for all $n$,

$$\min_{g \in B_\Gamma(n)\setminus\{e\}} d(g,e) \ge c\,|B_\Gamma(n)|^{-\beta}$$

(Aka et al., 2013). This expresses a polynomial lower bound on how close nontrivial words can approach the identity, tied to the growth rate of $\Gamma$; for nilpotent groups, $|B_\Gamma(n)|$ grows polynomially, by the Bass–Guivarc’h formula.

More generally, for connected real Lie groups $G$, one focuses on generic properties: for Haar-almost every $k$-tuple $(g_1,\dots,g_k)\in G^k$, the subgroup they generate is $\beta_k$-Diophantine for some exponent $\beta_k$ depending on $G$ and $k$.

2. Algebraic Obstructions and Extremality Criteria

Central to the study of Diophantine properties is the identification of algebraic obstructions via the notion of pencils $P_{W,r}$. For a matrix space $M_{m,n}(\mathbb{R})$, a pencil is defined by:

$P_{W,r} = \{ M \in M_{m,n}: \dim(MW) \le r \}$

where $W \le \mathbb{R}^{m+n}$ is a rational subspace. Dirichlet's principle yields that for any $M$ and $W$ with $\dim(W)=d$,

$$\beta(M) \ge \frac{d}{r} - 1 \quad\text{whenever}\quad \dim(MW)\le r$$

(Aka et al., 2014, Aka et al., 2016). Whenever $d/r-1 > n/m$ (the generic extremal value), containment in a pencil strictly forces $\beta(M)>n/m$ for all $M\in P_{W,r}$. The extremality criterion states that an analytic submanifold $S$ not contained in any constraining pencil attains the generic exponent $\beta(S)=n/m$ almost everywhere.

For submanifolds whose Zariski closure is defined over $\mathbb{Q}$, the almost-sure Diophantine exponent is rational and computed by maximizing $d/r-1$ over all rational pencils containing $S$:

$$\beta(S) = \max_{W \le \mathbb{Q}^{m+n},0 \le r \le m, S \subset P_{W,r}} \left( \frac{\dim W}{r} - 1 \right)$$

(Aka et al., 2014, Aka et al., 2016).

3. Diophantine Subgroups of Nilpotent and Solvable Lie Groups

Nilpotent Lie groups admit a robust characterization of Diophantine behavior in terms of the algebraic structure of their Lie algebra. Let $G$ be simply connected nilpotent of class $s$ with Lie algebra $\mathfrak{g}$. The ideal of laws $\mathcal{L}_{k,s}(\mathfrak{g})$—formal brackets vanishing identically—plays a critical role.

Aka–Breuillard–Rosenzweig–de Saxcé (Aka et al., 2013) establish that $G$ is Diophantine for $k$-tuples if and only if $\mathcal{L}_{k,s}(\mathfrak{g})$ is a Diophantine subspace of the free $s$-step nilpotent Lie algebra. Specifically, for rational nilpotent groups, the ideal is defined over a number field and hence automatically Diophantine. Explicit exponents $\beta_k$ can be computed by representation-theoretic data from the relatively free Lie algebra and are rational for large $k$ (Aka et al., 2014, Aka et al., 2016); for example, for the Heisenberg group $H_{2m+1}$,

$$\beta_k = 1 - \frac{1}{k} - \frac{2}{k^2} \quad \text{for } k \ge 2m$$

Solvable (but non-nilpotent) groups exhibit more subtle behavior. For the real affine group $\mathrm{Aff}(\mathbb{R})$, Varjú (Varjú, 2012) proves that in a one-parameter family, the set of parameters $\lambda$ for which the associated pair $(g_1(\lambda),g_2(\lambda))$ fails to be Diophantine has Hausdorff dimension zero—i.e., almost all such pairs are Diophantine.

Higher nilpotency class ($\geq 6$) or derived length ($\geq 3$) can lead to non-Diophantine examples, typically constructed via Liouville phenomena in the representation-theoretic decomposition of the free Lie algebra (Aka et al., 2013).

4. Diophantine Decision Problems in Algebraic and Reductive Groups

A fundamental development is the equivalence of Diophantine problems between isotropic reductive group schemes $G$ and their base rings $K$. For sufficiently isotropic $G$ (rank of root system $\geq 2$), both $G(K)$ and $K$ are existentially interpretable in each other, meaning that the Diophantine decision problem—deciding solvability of existential equations—is polynomial-time equivalent (Karp equivalent) between $G(K)$ and $K$ (Bunina et al., 2023, Voronetsky, 19 Apr 2025).

For Chevalley groups $G_\pi(\Phi,R)$ with indecomposable root system $\Phi$ of rank $\geq 2$ over a commutative ring $R$, Bunina–Myasnikov–Plotkin show that every one-parameter subgroup $X_\alpha$ is Diophantine, and the Diophantine problem in $G_\pi(\Phi,R)$ reduces to that in $R$. This result extends to various classes of rings and gives undecidability of the group-theoretic Hilbert’s tenth problem for $G_\pi(\Phi,\mathbb{Z})$.

5. Diophantine Conditions in Analysis of Differential Operators

Diophantine conditions appear as lower bounds on the symbol of left-invariant differential operators on compact Lie groups. For diagonal systems $P = (P_1,...,P_r)$, global solvability (GS) is equivalent to the existence of a uniform bound

$|\sigma^j_k(\xi)| \ge C \langle \xi \rangle^{-N}$

for nonzero diagonal entries of the symbol (Silva et al., 2024). Global hypoellipticity (GH) further requires that the set of representations where all symbol entries vanish is finite. These conditions generalize from diagonal to triangular systems with an additional uniform boundedness requirement on the dimensions of the irreducible representations.

Applied to products like $\mathbb{T}^n \times S^s$, the Diophantine property quantifies non-resonance conditions ensuring solvability and hypoellipticity for systems of vector fields and higher-order operators.

6. Intrinsic Diophantine Approximation on Simple Lie Groups

On simply-connected, almost-simple algebraic groups $G$, intrinsic Diophantine inequalities concern counting rational points of bounded height inside small neighborhoods of real points. Ghosh–Gorodnik–Nevo (Ghosh et al., 2021) establish asymptotic formulas:

$$N(\epsilon, T; x) = C(G)\,\epsilon^{\alpha(G)}\,T^{\beta(G)} + O(\epsilon^{\alpha(G)} T^{\beta(G) - \delta(G)})$$

where $\alpha(G) = \dim G$, $\beta(G)$ is the growth exponent of the adelic height-ball, and the error term $\delta(G)$ is explicit in terms of spectral gap data. These results generalize classical Diophantine approximation in Euclidean space to non-abelian groups, capturing group-dependent exponents and rates.

7. Consequences, Open Problems, and Further Directions

Key consequences include:

• Nilpotent Lie groups of class $\leq 5$ and metabelian groups are always Diophantine (Aka et al., 2013).
• Rational structure (defined over $\mathbb{Q}$) guarantees Diophantine exponents are rational and computable (Aka et al., 2014, Aka et al., 2016).
• The Diophantine decision problem for Chevalley and isotropic reductive groups is never simpler than for the base ring (Bunina et al., 2023, Voronetsky, 19 Apr 2025).

Major open questions persist regarding:

• Full Diophantine characterization for semisimple Lie groups—current best results only achieve subexponential lower bounds for non-algebraic generators.
• Quantitative rates for random walk equidistribution and almost-sure approximation in non-nilpotent, non-solvable cases.
• Structural description of the set of non-Diophantine tuples or parameters beyond dimension-zero/Hausdorff estimates.

This body of work forms a unified account of the Diophantine landscape for Lie groups, connecting algebraic, analytical, and arithmetic obstructions to explicit quantitative invariants.