Solomon's Criterion for Uniform Spreadness

• Uniform spreadness is defined by bounded displacement of Delone sets or strong group generating properties, characterized via spectral and probabilistic criteria.
• The criterion uses eigenvalue comparisons and fixed-point ratios to distinguish uniformly spread structures from irregular cases in tilings and group theory.
• Applications range from aperiodic substitution tilings to finite simple group generation, offering explicit thresholds and robust classification results.

Solomon's Criterion for Uniform Spreadness is a central quantitative framework used to detect when sets or structures generated by substitution and inflation processes exhibit uniform geometric or algebraic regularity analogous to lattices, or display group generation properties governed by probabilistic combinatorics. In contemporary mathematical literature, “uniform spreadness” refers to conditions under which a Delone set in Euclidean space (or a group in the context of finite group theory) admits a bounded displacement to a lattice, or possesses strong generating properties, with Solomon’s criterion providing explicit spectral, probabilistic, or group-theoretic thresholds. This concept has key impact both in the study of aperiodic tilings and in algebraic generation of finite simple groups.

1. Definitions and Preliminaries

The formalism of Solomon's criterion rests on foundational notions of Delone sets and uniform spread. A Delone set $\Lambda \subset \mathbb{R}^d$ is both uniformly discrete and relatively dense: there exist constants $0 < r \leq R < \infty$ so that for every $x \in \Lambda$, $B(x, r) \cap \Lambda = \{x\}$, and for every $y \in \mathbb{R}^d$, $B(y, R) \cap \Lambda \neq \emptyset$. Two Delone sets $\Lambda, \Gamma$ in $\mathbb{R}^d$ are said to be BD-equivalent (bounded displacement) if there is a bijection $\varphi: \Lambda \to \Gamma$ satisfying $\sup_{x \in \Lambda} \| x - \varphi(x) \| < \infty$. A Delone set is uniformly spread if it is BD-equivalent to some lattice in $\mathbb{R}^d$, equivalently to $c \cdot \mathbb{Z}^d$ for some $c > 0$. The Laczkovich theorem characterizes uniformly spread sets with an asymptotic density $d_\Lambda$ as those for which discrepancy ${\rm disc}_\Lambda(d_\Lambda; U)$ over bounded measurable regions $U$ is at most $C \cdot |\partial U|^{+(d-1)/d}$ for uniform $C < \infty$ (Smilansky, 28 Dec 2025).

In group-theoretic contexts, uniform spread is defined for a finite group $G$ as follows. $G$ has spread $k$ if for any $k$ non-identity elements $x_1, \dotsc, x_k$, there exists $g \in G$ with $\langle x_i, g \rangle = G$ for all $i$. $G$ has uniform spread $k$ if $g$ can be chosen from a single conjugacy class $s^G$. The uniform spread invariant $u(G)$ is the maximal $k$ such that $G$ has uniform spread $k$ (Harper, 2017).

2. Primitive Substitution Tilings and Spectral Matrices

Solomon’s eigenvalue criterion is applied within families of primitive substitution tilings. Consider a finite set of labelled prototiles $A = \{ T_1, \ldots, T_k \}$ in $\mathbb{R}^d$, each bi-Lipschitz to a closed ball. An inflation-substitution rule $\rho$ with expansion $\xi > 1$ acts by mapping each $T_j$ to finite patches of rescaled tiles in $\xi^{-1} A$. The substitution matrix $M_\rho$ records the number $a_{ij}$ of $\xi^{-1}T_i$ appearing in $\rho(T_j)$; primitiveness is ensured if some $M_\rho^p$ has all entries strictly positive (Smilansky, 28 Dec 2025).

By the Perron–Frobenius theorem, $M_\rho$ admits a unique dominant eigenvalue $\lambda_1 = \xi^d > 1$, with remaining eigenvalues $\lambda_2, \dotsc, \lambda_k$ of lesser modulus. Importantly, for each eigenvalue $\lambda$, the associated total eigenspace is $W_\lambda \subset \mathbb{C}^k$, with $1^\perp$ denoting the codimension-1 subspace orthogonal to the all-ones vector.

3. Solomon’s Spectral Criterion for Uniform Spreadness

Solomon’s criterion, as formalized by Smilansky, provides a dichotomy in terms of the substitution matrix spectrum. Let $\rho$ be a primitive substitution rule in $\mathbb{R}^d$, and $\Lambda$ a Delone set derived by selecting a control point per tile in a $\rho$-tiling. Let $\ell \geq 2$ be minimal such that $W_{\lambda_\ell} \not\subseteq 1^\perp$. Then (Smilansky, 28 Dec 2025):

• If $|\lambda_\ell| < \lambda_1^{(d-1)/d}$, then $\Lambda$ is uniformly spread.
• If $|\lambda_\ell| > \lambda_1^{(d-1)/d}$, then $\Lambda$ is not uniformly spread.
• If $|\lambda_\ell| = \lambda_1^{(d-1)/d}$, both cases may occur.

The proof relies on correlating the error term $O(|\lambda_\ell|^\ell)$ in counts of control points over regions formed by $\rho$-iteration with the boundary measure $|\partial U| \asymp \lambda_1^{(d-1)\ell/d}$. By Laczkovich’s discrepancy criterion, error bounded by boundary implies bounded displacement equivalence, giving the spectral threshold (Smilansky, 28 Dec 2025).

4. Application to $\alpha$-Kakutani Tilings of the Line

For commensurable $\alpha \in (0, 1/2]$ with $r_\alpha = (\log \alpha)/(\log(1-\alpha)) = n/m$ (gcd$(n, m)=1$), one constructs a 1-dimensional primitive substitution with expansion $\xi = \alpha^{-1/n}$ on $k = n+m-1$ prototiles. The substitution matrix $M_\alpha$ has characteristic polynomial $p_\alpha(x) = x^{n+m-1} - x^{m-1} - x^{n-1}$, where non-zero spectrum is given by the roots of $f_\alpha(x) = x^n - x^{n-m} - 1$. In dimension $d=1$, the Solomon criterion specializes: $\lambda_1 = \xi^1 > 1$, and the critical comparison is whether $|\lambda_2| < 1$ for the next eigenvalue with $W_{\lambda_2} \not\subseteq 1^\perp$ (Smilansky, 28 Dec 2025).

Uniform spreadness thus amounts to the strict inclusion of all non-unit eigenvalues of $M_\alpha$ in the open unit disk. Leverage of the classification of Pisot–Vijayaraghavan polynomials (Dubickas–Jankauskas 2014) determines that $|\lambda_2| < 1$ precisely for four minimal PV-polynomials

• $x^2 - x - 1$,
• $x^3 - x - 1$,
• $x^3 - x^2 - 1$,
• $x^4 - x^3 - 1$,

plus the trivial $r_\alpha=1$ case ($\alpha=1/2$). The permissible ratios are $r_\alpha \in \{1, 3/2, 2, 3, 4\}$ and numerically these correspond to $\alpha=1/2$, $\alpha=1/\varphi^2 \approx 0.381966$, $\alpha^2 = (1-\alpha)^3 \approx 0.43016$, $\alpha = (1-\alpha)^3 \approx 0.31767$, and $\alpha = (1-\alpha)^4 \approx 0.27551$. For all other $\alpha$, the criterion fails due to $|\lambda_2| > 1$ (Smilansky, 28 Dec 2025).

5. Probabilistic Solomon Criterion and Group Generation

In finite group theory, Solomon's probabilistic criterion—primarily as refined by Guralnick–Kantor and Burness–Guest—is employed to establish uniform spread properties of almost simple classical groups. Definitionally, given $s \in G$, let $M(G, s)$ be the set of maximal subgroups containing $s$. For any $x \in G$, the failure probability is $$P(x, s) = |\{g \in s^G : \langle x, g\rangle \neq G\}| / |s^G|$$. Key inequalities relate this to fixed-point ratios on $G/H$ coset actions. In applications,

• For each $x$ of prime order: $P(x, s) \leq \sum_{H \in M(G, s)} fpr(x, G/H)$, where $fpr(x, G/H) = \frac{|x^G \cap H|}{|x^G|}$.
• If for every $k$-tuple $(x_1, \dotsc, x_k)$, $\sum_{i=1}^k P(x_i, s) < 1$, then $G$ admits uniform spread $\geq k$ witnessed by $s^G$.

This framework underlies the establishment of lower bounds for $u(G)$ in families such as $PSp_{2m}(q)$ and $P\Omega_{2m+1}(q)$, with $u(G) \geq 2$ (except $S_6$), $u(G) \geq 4$ for $PSp_{2m}(q)$ when $q$ is odd and $m \geq 3$, and $u(G)$ diverging for large rank or large field except in bounded families (Harper, 2017).

6. Classification Theorems and Exceptional Sets

The ultimate classification result in the context of substitution tilings is as follows. Let $\Lambda_\alpha$ arise from an $\alpha$-Kakutani tiling of $\mathbb{R}$; then uniform spreadness obtains if and only if

$$r_\alpha = \frac{\log(\min\{\alpha, 1-\alpha\})}{\log(1 - \min\{\alpha, 1-\alpha\})} \in \{1, 3/2, 2, 3, 4\}$$

(Smilansky, 28 Dec 2025). In finite group generation, Harper’s theorems establish that for almost simple symplectic and orthogonal groups, $u(G)$ is typically unbounded except for specific small or “bad” families, and explicit lower and upper bounds are provided, unifying and strengthening prior work (Harper, 2017).

Context	Solomon's Criterion Formulation	Spectral/Probabilistic Threshold
Delone sets	Prim. subst. matrix eigenvalues	$|\lambda_\ell| \lessgtr \lambda_1^{(d-1)/d}$
Group Generation	Failure probabilities via fixed-point ratios	$\sum_{i=1}^k P(x_i, s) < 1$ for $k$-tuples

A plausible implication is that both spectral and probabilistic Solomon criteria serve as sharp demarcators between “exceptional” uniformly spread sets and general cases failing bounded displacement or uniform generation.

7. Open Problems and Extensions

Ongoing research is directed at classifying the full spectrum of substitution tilings and classical groups for which Solomon’s criterion ensures uniform spreadness. Open questions include exact determination of uniform spread invariants $u(G)$ across all classical and exceptional Lie-type families, extension to higher dimensional substitution tilings, and deeper structural understanding of the relationship between spectral gaps, discrepancy bounds, and uniform spread. Harper conjectures that unitary, even-dimensional orthogonal, and exceptional groups will behave analogously to symplectic and odd-orthogonal types with uniform spread diverging in large ranks, up to bounded exceptional cases (Harper, 2017). The spectral/fixed-point thresholds continue to guide the partition between uniformly spread and irregular structures.