Automorphic Subsets in Abelian Groups

• Automorphic subsets are F-subsets in finitely generated abelian groups defined via injective endomorphisms, capturing periodic behaviors through structured F-cycles and invariant subgroups.
• They utilize F-expansions and F-spanning sets to generalize base-k representations, ensuring that the underlying language of group elements is regular and automata-recognizable.
• These subsets underpin significant results such as the positive characteristic Mordell–Lang theorem, linking automata theory with arithmetic geometry and sparse language frameworks.

An automorphic subset—more precisely termed an F-subset in current research—is a well-structured class of subsets of a finitely generated abelian group $\Gamma$ equipped with an injective endomorphism $F\in\operatorname{End}(\Gamma)$. The theory generalizes the classical notions of $k$-automatic and $k$-normal subsets of $\mathbb{Z}$, extending them to the context of arbitrary finitely generated abelian groups and their endomorphisms, with foundational applications to problems such as the positive characteristic Mordell–Lang conjecture and the behavior of algebraic varieties over finite fields (Bell et al., 2017).

1. Definition and Structural Components

Given a finitely generated abelian group $\Gamma$ and injective $F\in\operatorname{End}(\Gamma)$, an F-subset of $\Gamma$ is any finite union of finite sums involving:

• Singletons $\{y\}\subseteq\Gamma$
• $F$-invariant subgroups $H\leq\Gamma$
• $F$-cycles of the form $$C(y;d)=\{y+F\,y+\cdots+F^{\ell-1}y:\ell\in\mathbb{N}\}$$, with $y\in\Gamma$, $d\in\mathbb{N}_+$.

This structure—introduced by Moosa and Scanlon—provides the correct abstraction for capturing sets characterized by automata and periodic behaviors, especially under algebraic or arithmetic operations.

2. F-Spanning Sets and F-Expansions

The key technical apparatus for analyzing F-subsets is the theory of F-expansions. A finite subset $E\subset\Gamma$ is called an F-spanning set if it satisfies a collection of axioms:

• $0\in E$ and $E$ is symmetric,
• $F(E)\subset E$,
• Every $x\in\Gamma$ possesses an "F-base" expansion $x=[x_0x_1\ldots x_m]_F=x_0+F x_1+\cdots+F^m x_m$ with $x_i\in E$,
• "Carry" and "borrow" properties ensuring well-behaved digit arithmetic.

This formalism generalizes the standard notion of base-$k$ expansions for integers, and underpins the encoding of group elements as words for finite automata processing (Bell et al., 2017).

3. F-Automatic Subsets and Regular Languages

A subset $S\subseteq\Gamma$ is $(E,F)$-automatic if the set of words $C=\{w\in E^*:[w]_F\in S\}$ forms a regular language, that is, it is recognized by a finite automaton over the alphabet $E$. This generalizes $k$-automatic sets (where $\Gamma=\mathbb{Z}$ and $F$ is multiplication by $k>1$), and is independent of the particular $F$-spanning set once existence is guaranteed.

The kernel characterization asserts that $S$ is $(E,F)$-automatic if and only if its $(E,F)$-kernel $$K(S)=\{S_u: u\in E^*, S_u=\{x\in\Gamma: [u]_F+F^k x\in S\}\}$$ is finite, mirroring the Myhill–Nerode theorem in automata theory.

Notion	Structure	Main Feature
F-spanning set	$E\subset\Gamma$	Digit set for expansions
F-cycle	$C(y;d)$	Periodic sum structure
F-automatic subset	$S\subset\Gamma$	Automaton-recognizable preimage

4. The Main Theorem: F-Subsets are F-Automatic

The principal result states that, assuming $F$ is injective and for each $d>0$ the operator $F^d-1$ is not a zero-divisor in $\mathbb{Z}[F]\subset\operatorname{End}(\Gamma)$, every F-subset of $\Gamma$ is F-automatic once an $F^r$-spanning set exists for some $r>0$. The class of F-automatic sets is closed under finite unions, finite sums, as well as under forming singletons and $F$-invariant subgroups.

It suffices to show F-cycles are F-automatic: for $C(y;d)$, the language underlying this set can be explicitly constructed and shown to be regular, thus F-automaticity follows (Bell et al., 2017).

5. F-Normality and Sparse Languages

The notion of F-normality generalizes Derksen’s concept of $p$-normal sets. A regular language $L\subset E^*$ is called sparse if its word-counting function grows polynomially; more precisely, $f_L(n)=|\{w\in L:|w|\leq n\}|=o(C^n)$ for all $C>1$. An F-sparse subset is any $S=\{[w]_{F^r}: w\in L\}$ for such a sparse $L$, some $r>0$, and F-normality is the property that $S$ is (modulo finite symmetric difference) a finite union of cosets $y+T+H$ with $y\in\Gamma$, $T$ F-sparse, and $H$ $F$-invariant.

The main result establishes that every F-subset is F-normal under the same preconditions as above (Bell et al., 2017).

6. Connections with Arithmetic Geometry and the Mordell–Lang Conjecture

A key application centers on the positive characteristic Mordell–Lang problem. Let $G$ be a semiabelian variety over a finite field $\mathbb{F}_q$ with $F$ the $q$-power Frobenius, and $\Gamma\leq G(K)$ an $F$-stable finitely generated subgroup. The isotrivial Mordell–Lang theorem asserts that $X\cap\Gamma$ is an F-subset of $\Gamma$ for any closed subvariety $X\subseteq G$. As a consequence, $X\cap\Gamma$ is both F-automatic and F-normal, extending results such as the Skolem–Mahler–Lech theorem to this context and asserting that the zero-set of a linear recurrence in characteristic $p$ is both $p$-automatic and $p$-normal (Bell et al., 2017).

7. Summary Table of Key Definitions

Term	Definition	Example
F-spanning set	Suitable $E\subset\Gamma$ for F-expansions	Digits for "base–F" expansion
F-automatic subset	Preimage forms a regular language via $w\rightarrow[w]_F$	$k$-automatic sets in $\mathbb{Z}$
F-cycle	$$C(y;d)=\{y+F y+\cdots+F^{\ell-1}y : \ell\in\mathbb{N}\}$$	Arithmetic progressions in $\mathbb{Z}$
F-normal set	Finite union of $y+T+H$, $T$ F-sparse, $H$ $F$-invariant	Satisfying sparse regular LLM

In summary, automorphic subsets, or F-subsets, capture key regularities in group-theoretic and algebraic contexts, unifying automata-theoretic and arithmetic concepts for groups equipped with endomorphisms, with applications in arithmetic geometry and the theory of regular languages (Bell et al., 2017).