Difference Bases for Cartesian Products

• Difference Bases for Cartesian Products are defined as subsets that represent every element as a difference of two fixed elements, extending classical additive theory.
• Explicit constructions using Galois rings and Teichmüller systems yield optimal bounds for g-difference bases in finite abelian groups.
• The categorical framework employing Cartesian difference categories and tangent bundle monads unifies discrete and continuous differentiation methods.

A difference basis for the Cartesian product of algebraic structures formalizes the concept of representing every element in the product as a difference of two elements from a fixed subset, possibly multiple times. Current research connects this notion, in both category theory and additive combinatorics, with advances in the structure of difference bases for finite abelian groups and the modeling power of Cartesian difference categories. The topic encompasses explicit construction results, asymptotic bounds, categorical interpretations via tangent bundle monads, and the generalization from continuous to discrete differentiation regimes.

1. Preliminaries: Difference Bases and Cartesian Products

A $g$-difference basis in a finite group $G$ is a subset $S \subseteq G$ such that for each $h \in G$, the equation $h = s_1 - s_2$ has at least $g$ solutions with $(s_1, s_2) \in S \times S$. For the Cartesian product $G^n$, a $g$-difference basis is similarly defined in terms of coordinate-wise differences. The minimal size of such a basis, denoted $\eta_{g}(G^n)$, is fundamental in additive combinatorics (Li et al., 28 Sep 2025). This concept generalizes to $g$-additive bases, wherein each element is represented by $g$ distinct sums.

2. Construction of Difference Bases for Products of Finite Abelian Groups

Explicit constructions for $g$-difference bases in $G^n$ often utilize algebraic objects such as Galois rings, Teichmüller systems, and relative difference sets. Key techniques include:

• When $G \cong \mathbb{Z}_{p^s}$, take the Galois ring $R = \mathrm{GR}(p^s,n)$ with additive group structure matching $G^n$.
• For $R \times R \cong G^{2n}$, form slices $S_i = \{(x, \alpha_i x): x \in R^*\}$ for selected $\alpha_i$ from a Teichmüller system $T \subset R$, ensuring $\alpha_i - \alpha_j$ are units (Li et al., 28 Sep 2025).
• The union $S = \bigcup_{i=1}^k S_i$ gives a $(k^2 - k)$-difference basis, with $|S| \leq k(|R| - 1)$.

For $g = k^2 - k$, the bound $\eta_{g}(G^{2n}) \leq k (p^{sn} - 1)$ holds. For small $g$, specializing $k$ yields:

$g$	$\eta_{g}(G^{2n})$, $\nu_{g}(G^{2n})$
$1,2$	$\leq 2(p^{sn} - 1)$
$3,4,5,6$	$\leq 3(p^{sn} - 1)$

Direct product and quotient constructions allow extension to higher $n$ and refine bounds via combinatorial design-theoretic techniques (Li et al., 28 Sep 2025).

3. Asymptotic Behavior and Optimality

For groups with suitably large invariant factors (typically excluding very small groups such as $\mathbb{Z}_2$):

• The minimal size of a $g$-difference basis in $G^n$ is asymptotically $(\sqrt{g} + O(1))|G|^{n/2}$ as $n \to \infty$.
• For $g=1$, one obtains $\eta_1(G^n) \sim |G|^{n/2}$, matching counting lower bounds.
• Tight asymptotics are obtained via product constructions and leveraging the algebraic non-degeneracy from Galois rings and difference set selection.

4. Categorical Framework: Cartesian Difference Categories

Cartesian difference categories generalize Cartesian differential categories by introducing an infinitesimal extension operator $\varepsilon$ and a difference combinator $[f]$. The formal definition requires:

• Underlying left-additive Cartesian category structure;
• $\varepsilon: \mathbb{X}(A,B) \to \mathbb{X}(A,B)$, a monoid morphism;
• $[f]: A \times A \to B$ for each arrow $f$;
• Modified axioms, including

$$f(x + \varepsilon(y)) = f(x) + \varepsilon([f](x,y)),$$

$$[f](x, y + z) = [f](x, y) + [f](x + \varepsilon(y), z),$$

and a chain rule for composition.

In finite difference calculus, $\varepsilon$ is often the identity and $[f](x, y) = f(x+y) - f(x)$. The relaxation of strict linearity (additivity holds only "up to translation" by $\varepsilon$) enables modeling of discrete difference bases (2002.01091, Alvarez-Picallo et al., 2020).

The tangent bundle monad construction on $A$,

$$T(A) = A \times A, \quad T(f) = \langle f \circ \pi_0, [f] \rangle,$$

and associated monad structure (unit $\eta_A$, multiplication $\mu_A$), organizes the theory of difference bases for products, as the second coordinate provides a canonical difference basis.

5. Relationship to Change Action Models and Applications

Cartesian difference categories serve as a bridge between the classical (smooth) regime of differential categories and the more general, discrete change action models:

• Every Cartesian differential category is a Cartesian difference category (with $\varepsilon = 0$).
• Many change action models, capturing incremental/discrete differentiation, are subsumed by the difference framework when well-behaved (2002.01091, Alvarez-Picallo et al., 2020).
• The categorical formalism supports applications to computer science (incremental computation semantics), combinatorics, and higher-order difference calculus on product spaces.

Difference bases thus have both explicit combinatorial constructions for finite abelian groups and categorical models accommodating both smooth and discrete differentiation paradigms.

6. Implications, Limitations, and Extensions

The existence of difference bases for Cartesian products informs the decomposition of change across coordinates and underlies efficient descriptions of product group behavior. For very small groups, adjustments to the bounds and conditions are necessary; admissibility is crucial in 2-group settings for the effectiveness of Galois ring constructions.

The theory offers unification of notions previously treated separately: classical bases for smooth functions, finite difference bases for discrete settings, and abstract difference bases for general product objects in additive categories. The tangent bundle monad and Kleisli category stability deepen the algebraic and categorical understanding.

Future avenues include refinement of bounds in special cases, deeper analysis of categorical difference structures with more exotic infinitesimal extensions, and applications to program semantics and automatic differentiation where both product structure and difference bases are crucial.