Expansivity Modulo an Ideal

• Expansivity modulo an ideal is a separation property where distinct points are separated by actions outside ideal-negligible subsets, extending classical dynamical concepts.
• In uniform transformation semigroups and ring theory, it is characterized via expansive indices and ideal generators, linking topological dynamics with algebraic structure.
• Its applications in arithmetic distance geometry lead to modular rigidity results, establishing sharp bounds on geometric configurations through congruence conditions.

Expansivity modulo an ideal denotes a family of separation phenomena in which an ideal determines the exceptional sets on which separation may fail, or the ideal-theoretic framework through which expansivity is analyzed. In uniform transformation semigroups, the notion is formalized by requiring that distinct points be separated by some action parameter outside every ideal-small subset of the acting semigroup (Shirazi et al., 24 Aug 2025). In commutative algebra, expansivity is translated from open-cover dynamics to generators of ideals and their refinements, and passage to quotients by invariant ideals gives a precise ideal-sensitive descent principle (Artigue et al., 2018). In arithmetic distance geometry, the expression appears as an interpretation of rigidity for Euclidean configurations whose squared distances occupy only a small number of residue classes modulo a prime ideal, yielding sharp upper bounds on the size of the configuration (Nozaki, 2022, Nozaki, 2023).

1. Formal definition in uniform transformation semigroups

Let $(T,(X,\mathcal{K}))$ be a uniform transformation semigroup, where $T$ is a semigroup, $X$ is the phase space, $\mathcal{K}$ is a compatible uniform structure on $X$, and $\mathcal{I}$ is an ideal on $T$. The system is expansive modulo an ideal $\mathcal{I}$, or $\mathcal{I}$-expansive, if there exists an entourage $\alpha \in \mathcal{K}$ such that

$T$0

The entourage $T$1 is the expansive index. An equivalent formulation is that for all distinct $T$2, the separator set

$T$3

does not belong to $T$4, where $T$5 is an $T$6-expansive index (Shirazi et al., 24 Aug 2025).

This definition interpolates between classical expansivity and weaker ideal-constrained variants. The classical notion is recovered when $T$7. More generally, the ideal specifies which subsets of time or action parameters are considered negligible. If $T$8 is $T$9-expansive, then it is also $X$0-expansive for any ideal $X$1. Thus smaller ideals impose stronger separation requirements, while larger ideals weaken them.

2. Comparison with classical expansivity

The ideal-modified definition relaxes the classical requirement that distinct points must separate at some time without exception. Here separation is required only outside sets belonging to the chosen ideal. The resulting hierarchy is naturally interpreted as a scale of admissible exceptional sets.

Ideal $X$2	Property	Interpretation
$X$3	Classical expansivity	Some time $X$4 separates $X$5 and $X$6
Finite sets	Finite-exception expansivity	Separation occurs for all but finitely many $X$7
Arbitrary ideal	$X$8-expansivity	Separation occurs except perhaps on an ideal-small set

The converse relation with classical expansivity fails in general. A counterexample is given by $X$9 with the usual metric, $\mathcal{K}$0 where $\mathcal{K}$1, and the ideal $\mathcal{K}$2. This semigroup is classically expansive, but it is not $\mathcal{K}$3-expansive. Accordingly, $\mathcal{K}$4-expansivity is not equivalent to classical expansivity, and the choice of ideal is mathematically substantive rather than merely notational (Shirazi et al., 24 Aug 2025).

3. Ring-theoretic formulation and ideal generators

For commutative rings with identity, expansivity is recast in purely algebraic terms. A finite set $\mathcal{K}$5 of ideals is a generator if

$\mathcal{K}$6

The analogue of cover refinement is the relation $\mathcal{K}$7, meaning that for every $\mathcal{K}$8, there is $\mathcal{K}$9 such that $X$0. If $X$1 is a ring automorphism, then $X$2 is expansive if there exists a generator $X$3 such that for any generator $X$4, there is $X$5 with

$X$6

It is positively expansive if the same condition holds with

$X$7

A generator is $X$8-minimal if $X$9 for every generator $\mathcal{I}$0, and the existence of such a generator is called 0-expansivity (Artigue et al., 2018).

This framework is an algebraic translation of topological dynamics. For a compact space $\mathcal{I}$1, the ring $\mathcal{I}$2 of real-valued continuous functions reflects the topology of $\mathcal{I}$3 through its ideals: open subsets correspond to ideals, and open covers correspond to generators. In this setting, the algebraic definition recovers the classical one: a homeomorphism $\mathcal{I}$4 on $\mathcal{I}$5 is expansive if and only if the induced automorphism on $\mathcal{I}$6 is expansive.

4. Structural consequences in commutative algebra

The ring-theoretic theory yields strong structural characterizations. A ring $\mathcal{I}$7 admits a $\mathcal{I}$8-minimal generator if and only if it is a finite product of local rings. In that case, $\mathcal{I}$9 has finitely many maximal ideals, and the minimal generator can be taken as a family $T$0 in which each $T$1 is idempotent and principal and the ideals are pairwise orthogonal, $T$2 for $T$3. If $T$4 admits a positively expansive automorphism, then $T$5 has finitely many maximal ideals. For a principal ideal domain, the following are equivalent: $T$6 admits a positive expansive automorphism, the identity automorphism on $T$7 is expansive, and $T$8 has finitely many maximal ideals (Artigue et al., 2018).

The ideal-sensitive aspect is especially clear in the quotient construction. If $T$9 is expansive or positively expansive and $\mathcal{I}$0 is an $\mathcal{I}$1-invariant ideal, then the induced automorphism on $\mathcal{I}$2 is also expansive or positively expansive. This is the ring-theoretic version of passing to dynamics “modulo an ideal.” The same work shows that algebraic expansivity is stronger than topological expansivity on $\mathcal{I}$3 with the Zariski topology: if $\mathcal{I}$4 is algebraically expansive, then the induced map on $\mathcal{I}$5 is topologically expansive, but the converse fails. The ring $\mathcal{I}$6, a subring of the rationals with denominators not divisible by $\mathcal{I}$7 or $\mathcal{I}$8, has two maximal ideals but no $\mathcal{I}$9-minimal generator; its identity automorphism is positively expansive but not $\mathcal{I}$0-expansive. By contrast, any local ring has $\mathcal{I}$1 as a $\mathcal{I}$2-minimal generator, and finite rings or finite products of local rings always admit minimal generators.

5. Shadowing, uniqueness, and topological stability

Within compact Hausdorff transformation semigroups, expansivity modulo an ideal interacts decisively with shadowing modulo the same ideal. A key uniqueness statement asserts that if a system is $\mathcal{I}$3-expansive with index $\mathcal{I}$4, and $\mathcal{I}$5 is an entourage satisfying $\mathcal{I}$6, then any two $\mathcal{I}$7-traces of the same pseudo-orbit must be identical. In other words, $\mathcal{I}$8-expansivity enforces uniqueness of shadowing points for ideal-constrained pseudo-orbits (Shirazi et al., 24 Aug 2025).

The principal stability theorem states that if a compact Hausdorff transformation semigroup has the shadowing property modulo $\mathcal{I}$9 and is expansive modulo $\alpha \in \mathcal{K}$0, then it is topologically stable modulo $\alpha \in \mathcal{K}$1. This extends classical stability theorems, including Walters-type results, to ideal-constrained dynamics. The same framework also examines the relationship between shadowing modulo an ideal and the conventional shadowing property. In this sense, expansivity modulo an ideal functions as one half of a rigidity pair: separation controls ambiguity, while shadowing controls persistence under perturbation.

6. Arithmetic-geometric rigidity for few residue classes

A different but closely related use of the language appears in Euclidean distance geometry over number fields. Let $\alpha \in \mathcal{K}$2 be the ring of integers of an algebraic number field $\alpha \in \mathcal{K}$3 embedded into $\alpha \in \mathcal{K}$4, let $\alpha \in \mathcal{K}$5, and let

$\alpha \in \mathcal{K}$6

be the set of squared distances. If $\alpha \in \mathcal{K}$7 and there exist $\alpha \in \mathcal{K}$8 values $\alpha \in \mathcal{K}$9, distinct modulo a prime ideal $T$00, each nonzero modulo $T$01, such that every element of $T$02 is congruent to one of the $T$03, then

$T$04

The same bound holds in the localized setting $T$05. This is Theorem 3.1 and its corollary in the local and integral formulations (Nozaki, 2022).

The proof uses the polynomial method in the form of Koornwinder’s method. To each $T$06 one associates

$T$07

which lies in $T$08, the space of real polynomials of degree at most $T$09 in $T$10 variables. Under the modular hypothesis, the evaluation matrix of these polynomials on the points of $T$11 is congruent modulo $T$12 to a diagonal matrix with diagonal entries that are units. Nakayama’s Lemma then yields linear independence of $T$13, and therefore

$T$14

In the discussion of the paper, this is presented as “Expansivity/Rigidity Modulo an Ideal.” Many actual Euclidean distances may collapse to a single residue modulo $T$15, yet the spread of the configuration, as measured by $T$16, remains tightly constrained. The same discussion notes that this is analogous to Frankl-Wilson type “modular rigidity” in subset intersection theory. The paper also records an example attaining the bound: $T$17 for which $T$18, $T$19, $T$20, $T$21, and $T$22.

7. The one-distance case modulo a prime ideal

The case in which all nonzero squared distances are congruent to a single residue class is a particularly rigid extremal regime. Let $T$23 be a number field embedded into $T$24, let $T$25, let $T$26 be a prime ideal, and let $T$27 be the localization at $T$28. If the squared distances of a finite set $T$29 lie in $T$30 and each squared distance is congruent to some constant $T$31, then

$T$32

This generalizes the classical statement for odd integral squared distances, where the residue field is $T$33 (Nozaki, 2023).

The attainability of the bound depends exactly on the characteristic of the residue field. If $T$34 has characteristic $T$35, then there exists $T$36 with $T$37 and all squared distances congruent modulo $T$38 if and only if

$T$39

If the residue characteristic is an odd prime $T$40, then there exists such a set if and only if

$T$41

Theorem 3.1 gives the necessity $T$42, Theorem 3.2 gives sufficiency for odd $T$43, and Theorem 3.4 gives the characteristic-$T$44 criterion. Examples attaining the upper bound include the regular simplex in $T$45 together with its centroid when $T$46, and an explicit family

$T$47

in the odd-characteristic case.

The presentation explicitly interprets these results in terms of modular rigidity and combinatorial expansivity modulo an ideal. The maximum cardinality reflects the fact that a local congruence condition on distances forces a very tight combinatorial structure, essentially a simplex-plus-center configuration, and only for specific congruence classes of $T$48 can the upper bound be attained. The same work also notes connections with $T$49-distance sets, the Larman-Rogers-Seidel ratio, and the construction of further modular $T$50-distance sets by lifting to extensions.