(joint work with Yuyan He)

1. Introduction and Historical Context

The sum-product phenomenon is a core theme in additive combinatorics, asserting that a subset \(A\) of a ring or field cannot be simultaneously close to being an additive subgroup and a multiplicative subgroup. In other words, both the sumset \(A+A\) and the product set \(AA\) cannot remain small unless \(A\) itself structurally mirrors an existing algebraic subfield.

This behavior was first classically formalized for finite sets of integers, establishing that there exist constants \(C, \delta > 0\) such that

\[ \max\{|A+A|,\;|AA|\} \ge C|A|^{1 + \delta}. \]

In fields of finite characteristic, the landscape is inherently more intricate due to the presence of actual subfields. Classical extensions of this phenomenon to finite fields demonstrate that if a subset \(A\) does not structurally mimic a subfield, it must exhibit significant expansion under either addition or multiplication.

1.1. The Model-Theoretic Approach

This talk introduces a model-theoretic generalization of the sum-product phenomenon. Instead of quantifying the "size" of a set using traditional counting measures or discrete cardinalities, we transition to a model-theoretic framework where size is governed by a continuous real dimension encoded within first-order logic. We demonstrate that small expansion under addition and multiplication implies the presence of an explicitly definable subfield of comparable dimension.

2. Axiomatizing Continuous Real Dimension

Let \(T\) be a complete first-order theory and \(M \models T\) a model. A continuous real dimension is a mapping \(\delta: Def(M) \to \mathbb{R}_{\ge 0}\cup\{+\infty\}\) on the collection of parameter-definable sets satisfying the following axioms for any \(X,Y \in Def(M)\):

 

• Union: \(\delta(X\cup Y)=\max\{\delta(X),\delta(Y)\}\)

 

• Product: \(\delta(X\times Y) = \delta(X) + \delta(Y)\)

 

• Finiteness: \(\delta(X) = 0\) if \(X\) is a finite set

 

• Intersection: \(\delta(X\cap Y)\leq \min\{\delta(X),\delta(Y)\}\)

 

• Invariance: \(\delta(X) = \delta(Y)\) if there exists a definable bijection \(f:X \to Y\)

 

• Subadditivity: for any definable surjection \(f: X \to Y\), \[ \delta(X) = \sup\Bigl\{ \alpha+\beta\; :
  \alpha\in\mathbb{R}\cup\{\infty\}, \; \beta \leq\bigl\{ z\in Y\;:\;\delta(f^{-1}(z))\geq\alpha\bigr\} \Bigr\}. \]

 

• Continuity: for any formula \(\phi(\bar x,\bar y)\) and \(s < t \in \mathbb{R}\), there is a \(\varnothing\)-definable \(D\) such that \[ \{a\;:\;\delta(\phi(\bar x,a))\leq s\} \subseteq D \subseteq \{b\;:\;\delta(\phi(\bar x,b)) < t\}. \]

 

This dimension generalizes classical geometric dimensions; for instance, o-minimal and geometric theories using algebraic closure (acl) dimension, alongside some theories of finite Morley rank. Importantly, it captures coarse pseudofinite dimension, which treats asymptotic bounds of finite sets within an ultraproduct as a real-valued dimension.

Using this framework, we extend \(\delta\) to partial types by taking the infimum over defining formulas, which allows us to naturally recover robust model-theoretic definitions for \textbf{genericity} and \textbf{independence} \((\vec a \perp_A \vec b)\).

3. Combinatorial Machinery in the Dimensional Setting

We adopt some tools of classical additive combinatorics into continuous dimensions.

3.1. Generalized Ruzsa Triangle Inequality

We prove that for type-definable sets \(X, Y, Z\) of finite dimension in a dimensional group, the following holds:

\[ \delta(X) + \delta(Y-Z) \leq \delta(X-Y) + \delta(X-Z).\]

The proof relies on generic independent realizations, the underlying symmetry and additivity of the dimension function.

3.2. Plünnecke-Ruzsa Sumset Estimates

We adapt Plünnecke's original strategy of utilizing commutative layered graphs. By tracking injections within a definably commutative three-layered graph \(\Gamma=(U_0 \sqcup U_1 \sqcup U_2, E)\), we establish that if a type-definable set expands minimally at the first layer, its higher expansions are bounded. This yields the Sumset Estimate: if \(\delta(X + Y) = \delta(X) < \infty\), then for any \(m, n \in \mathbb{N}\), \(\delta(mY - nY)\le \delta(X)\).

3.3. Balog–Szemerédi–Gowers (BSG) Lemma and the Katz–Tao Theorem

Using \(n\)-gons and triangles we prove a weak variant of the BSG lemma. This serves as the bridge to proving the dimensional version of the \textbf{Katz–Tao lemma}. We show that if a type-definable set \(X\) has identical sum and product dimensions (\(0 < \delta(X) = \delta(X+X) = \delta(XX) <\infty\)), then for any generic types \(p, q\) on \(X\), the difference of their products expands minimally:

\[ \delta(pq - pq) = \delta(X). \]

4. The Main Theorem: Constructing the Definable Field

Theorem 1. Let \(F\) be a sufficiently saturated model of a theory expanding the theory of fields, equipped with a continuous real dimension \(\delta\). Let \(X\) be a complete type over a small parameter set \(A\) satisfying \(0 < \delta(X+X) = \delta(XX) = \delta(X) < \infty\). Let \(p\) be a generic type on \(X\) over \(A\), and define \(Y = a^{-1}p\) for some \(a \models p\). Then \(\frac{Y-Y}{Y-Y}\) is a definable field, and \(\delta\;\left(\frac{Y-Y}{Y-Y}\right) = \delta(X)\).

4.1. Outline of Proof Strategy

 

1. The \((\star)\) Property:  We define a stabilizing property for any element \(x\in F\): \(x\) satisfies \((\star)\) if \(\delta(xY + Y) = \delta(Y)\). Using the dimensional Ruzsa triangle inequality, we prove that the collection of elements satisfying \((\star)\) forms a field.

 

2. Equivalence to the Quotient Set: We show via a non-injectivity argument that an element \(x\) satisfies \((\star)\) if and only if it can be written as a quotient of differences: \(x \in \frac{Y-Y}{Y-Y}\).

 

3. Definability via Subfield Stabilization Theorems: While the quotient set is initially type-definable, model-theoretic structural properties of type-definable subfields of finite dimension inside a dimensional field ensure that it is automatically first-order definable over the same parameters.

 

5. Finitary Applications and Conclusion

The advantage of this model-theoretic approach is its generalizability across arbitrary fields. By applying our main theorem to coarse pseudofinite dimension and invoking ultraproduct structures, we immediately recover a clean, general finitary statement across all fields.

Corollary 1. For any \(\delta \in (0, 1)\), there exist \(\epsilon(\delta)\in (0, 1)\) and \(N(\delta) \in \mathbb{N}\) such that for any finite subset \(A\subseteq F\) of a field, if \(|A+A|,\,|AA| \leq |A|^{1+\epsilon(\delta)}\) and \(|A|\geq N(\delta)\), then there is a subfield \(E\subseteq F\) satisfying

\[ |A|^{1-\delta} \leq |E| \le |A|^{1+\delta}. \]