# P3002: Erdős's sum-product growth problem for finite sets of integers

- ID: `P3002`
- Reference: `erdos-problem-52`
- Page: https://theoremdb.org/statements/P3002
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
\[
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
\]
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$.

### Context

This problem asks whether the sumset and product set of a finite set of integers cannot both be simultaneously too small, specifically whether at least one of them must grow almost as fast as the square of the set's cardinality.

### Problem setup

- **Definition (For a finite set $A$ of integers, the sumset $A+A$).** For a finite set $A$ of integers, the sumset $A+A$ is the set $\{a+b : a,b \in A\}$.
- **Definition (For a finite set $A$ of integers, the product set $A \cdot A$).** For a finite set $A$ of integers, the product set $A \cdot A$ is the set $\{a \cdot b : a,b \in A\}$.
- **Definition (For a finite set $S$, the notation $|S|$).** For a finite set $S$, the notation $|S|$ denotes the cardinality of $S$.
- **Definition (The notation $\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}$).** The notation $\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}$ means that the larger of the two cardinalities $|A+A|$ and $|A \cdot A|$ is at least $C \cdot |A|^{2-\varepsilon}$.
- **Remark.** This problem asks whether the sumset and product set of a finite set of integers cannot both be simultaneously too small, specifically whether at least one of them must grow almost as fast as the square of the set's cardinality.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
\[
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
\]
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 52 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
\[
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
\]
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 52 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
\[
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
\]
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 52 as open. The unresolved remainder is the full displayed statement.

A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
\[
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
\]
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 52 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 52 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
- The pinned Formal Conjectures declaration was matched by problem number and source locator.
- The controlled TheoremDB source corpus was checked for an already published record with the same slug.

### Open directions

- **Route 1** (reported): Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that \[ \max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon} \] holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ denotes the product set $\{a \cdot b : a,b \in A\}$. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-52`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 52, maintained status record. Erdős Problems record 52, checked 2026-08-01. Problem 52; status field and linked bibliography https://www.erdosproblems.com/52
   - Also cited at Problem 52; status snapshot 8138974387d9030542daabe67faaa33eff9356f8
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Erdős's sum-product growth problem for finite sets of integers: Records the current open status and links the literature attached to this exact numbered problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 52 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Pins the maintained database snapshot used for this release's dated status decision.
   - Source used to assess the problem's recorded status.
   - For Erdős's sum-product growth problem for finite sets of integers: Pins the maintained database snapshot used for this release's dated status decision.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 52. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/52.lean:L34; theorem erdos_52; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/52.lean#L34
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source use: original_summary
   - Supplies the pinned formal declaration whose human-readable editorial statement is published here.
   - Source used to assess the problem's recorded status.
   - For Erdős's sum-product growth problem for finite sets of integers: Supplies the pinned formal declaration whose human-readable editorial statement is published here.