Frostman random variables, entropy inequalities, and applications

1. Problem Statement and Motivation

The paper addresses the discretized sum-product phenomenon in the context of dependent random variables and general polynomial maps. The classical Erdős–Szemerédi sum-product conjecture posits that a finite set $A$ cannot be simultaneously additively and multiplicatively structured (i.e., $|A+A|$ and $|A \cdot A|$ cannot both be small).

Recent work by M´ath´e and O'Regan extended this to entropy for independent Frostman random variables, showing that for independent $(s, C)$-Frostman variables $X, Y$, $\max\{H_n(X+Y), H_n(XY)\} \ge n(s+\epsilon)$.

The Core Problem: This paper seeks to generalize these results to:

1. Dependent variables: Where $X$ and $Y$ are not independent.
2. General polynomials: Instead of just $x+y$ and $xy$, considering $\phi(X, Y)$ where $\phi$ is a polynomial of degree $d \ge 2$ derived from a diagonal form via a rational linear change of variables.
3. Frostman Hierarchy: Establishing how different levels of "non-concentration" (Joint, Marginal, Conditional) affect entropy expansion bounds.

2. Methodology and Novel Framework

The authors introduce a novel multi-step entropy framework that combines tools from geometric measure theory, additive combinatorics, and information theory.

A. The Frostman Hierarchy for Bivariate Variables

The paper defines a hierarchy of non-concentration conditions for a bivariate random variable $(X, Y)$ with dimension parameters $s_1, s_2$:

1. Jointly Frostman: $P(X \in B(x, r_1), Y \in B(y, r_2)) \le C r_1^{s_1} r_2^{s_2}$.
2. Conditionally Frostman: $P(X \in B(x, r) | Y \in E) \le C r^s P(Y \in E)$.
3. Marginally Frostman: $X$ and $Y$ are individually Frostman.

The authors prove that these conditions form a strict hierarchy:
$$\text{Independent \& Marginal} \implies \text{Conditional} \implies \text{Joint} \implies \text{Marginal}$$
Crucially, they show that different levels of the hierarchy yield different entropy lower bounds, necessitating distinct proof strategies.

B. The Five-Step Proof Strategy

The proof of the main entropy theorems proceeds through five distinct stages:

1. Distance Entropy Lower Bounds:

   • Case $s > 1/2$: Uses Mattila's Fourier-analytic estimates on distance sets. If $W$ is a planar $(2s, C)$-Frostman variable, the entropy of the distance $|W - W'|$ (where $W'$ is an i.i.d. copy) is bounded below by $n(s + \epsilon)$.
   • Case $s \le 1/2$: Standard distance estimates fail. The authors adapt the robust projection and thin-tube framework of Shmerkin and Wang [35]. This requires $X$ and $Y$ to be independent (product structure) to establish a distance entropy gain.

2. Diagonal Quadratics (Independent Case):

   • For $\phi(x, y) = \alpha x^2 + \beta y^2$, the authors relate the distance $|W_1 - W_2|^2$ to the sum $X+Y$ and the quadratic form using polarization identities.
   • They apply discretized submodularity and Plünnecke–Ruzsa type inequalities to show that if the distance entropy is high, then $\max\{H_n(X+Y), H_n(\alpha X^2 + \beta Y^2)\}$ must also be high.

3. Diagonal Polynomials (Inductive Reduction):

   • For higher-degree diagonal polynomials $\rho_1(x) + \rho_2(y)$, the authors use a two-point differencing scheme.
   • By conditioning on dyadic differences $D_n(X_1 - X_2)$, they reduce the degree of the polynomials from $d$ to $d-1$.
   • This relies on the local weak bi-Lipschitz decomposition of polynomials away from critical points. The induction closes with the quadratic base case.

4. Dependent Variables (Balog–Szemerédi–Gowers Mechanism):

   • To handle dependent variables, the authors employ a discretized entropy Balog–Szemerédi–Gowers (BSG) theorem.
   • This mechanism transforms a dependent pair $(X, Y)$ into a family of conditionally independent pairs $(X'', Y'')$ given discretized variables $D_n(X'), D_n(Y')$.
   • This step introduces a "dependence penalty" measured by the discretized mutual information $I_n(X; Y) = H_n(X) + H_n(Y) - H_n(X, Y)$. This explains why the bound for jointly Frostman variables includes the term $\frac{H_n(X) + H_n(Y)}{2}$ rather than just the max.

5. General Polynomials (Rational Linear Changes):

   • For general forms $\phi(x, y) = \rho_1(\xi_{11}x + \xi_{12}y) + \rho_2(\xi_{21}x + \xi_{22}y)$, the authors use quantitative Ruzsa-type entropy inequalities for rational coefficients.
   • They prove that under conditional Frostman assumptions, the joint entropy $H_n(X, Y)$ is comparable to $H_n(X+Y, \phi(X, Y))$ via a quantitative local invertibility argument (using the Jacobian determinant and Lojasiewicz inequalities). This allows them to eliminate the joint entropy term from the final bound.

3. Key Results

Theorem 1.2 (Jointly Frostman, Diagonal Polynomials)

Let $(X, Y)$ be jointly $(s, s, C)$-Frostman. Then for diagonal polynomials $\rho_1, \rho_2$:
$$\max \left\{ \frac{H_n(X) + H_n(Y)}{2}, H_n(X+Y), H_n(\rho_1(X) + \rho_2(Y)) \right\} \ge n(s + \epsilon)$$
Significance: The inclusion of $\frac{H_n(X) + H_n(Y)}{2}$ is necessary when variables are dependent but only jointly Frostman.

Theorem 1.3 (Conditionally Frostman, General Polynomials)

Let $(X, Y)$ be conditionally $(s, C; s, C)$-Frostman. Let $\phi$ be a general polynomial form (diagonal or non-diagonal) derived from a rational linear change of variables. Then:
$$\max \{ H_n(X+Y), H_n(\phi(X, Y)) \} \ge n(s + \epsilon)$$
Significance: Under the stronger conditional hypothesis, the baseline term $\frac{H_n(X) + H_n(Y)}{2}$ is no longer needed, and the result holds for general polynomials.

Discretized Sum-Product Estimates (Theorems 1.4–1.6)

The authors translate these entropy results into geometric combinatorics for $\delta$-separated sets $A \subset [0, 1]$ with cardinality $\approx \delta^{-s}$.

• Theorem 1.4: For a dense graph $G \subseteq A \times A$ (size $\ge \delta^\epsilon |A|^2$), the sumset and polynomial image cannot both be small:
  $$E_\delta(A +_G A) + E_\delta(\phi_G(A, A)) \gg \delta^{-\epsilon} |A|$$
• Theorem 1.5: Extends this to graphs with minimum degree conditions (two-sided density).
• Theorem 1.6: Establishes stability under taking subsets, showing that any large subset $B \subseteq A$ also exhibits expansion.

4. Significance and Applications

1. Resolution of Dependence: The paper successfully extends sum-product phenomena to dependent random variables, a significant hurdle in previous literature which largely relied on independence.
2. Frostman Hierarchy: It clarifies the precise relationship between the strength of the non-concentration hypothesis (Joint vs. Conditional) and the resulting entropy gain.
3. General Polynomials: It moves beyond the specific case of $x+y$ and $xy$ to arbitrary polynomials of degree $d \ge 2$, including non-diagonal forms.
4. Adaptable Sets: The results provide new bounds for the Erdős–Szemerédi sum-product conjecture in the setting of "adaptable sets" (sets with controlled energy), offering a continuous-to-discrete bridge that yields results for general graphs, not just complete graphs.
5. Technical Innovation: The combination of Shmerkin–Wang's thin-tube arguments (for $s \le 1/2$) with entropy-based BSG mechanisms creates a robust toolkit for analyzing discretized entropy in dependent settings.

5. Sharpness and Examples

The authors provide a series of examples (Section 8) demonstrating that:

• The term $\frac{H_n(X) + H_n(Y)}{2}$ in Theorem 1.2 cannot be removed for jointly Frostman variables.
• The condition of conditional Frostman is necessary for the clean bound in Theorem 1.3; dropping it allows for counterexamples where both sums and products are small.
• The requirement that the polynomial $\phi$ cannot be reduced to a univariate function is essential.
• The dependence on the Frostman constant $C$ (via $\log C$ terms) is necessary when $C$ grows with the scale $n$.

In summary, this paper represents a major advancement in the field of additive combinatorics and fractal geometry, providing a unified and rigorous framework for understanding entropy expansion in dependent, non-concentrated systems.