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

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 $\{a \cdot b : a,b \in A\}$.

A finite mathematical diagram showing one finite set feeding its sumset and product set. — One finite set feeding its sumset and product set.

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Definitions and notation

1Context

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.

2Definitions

Definition 1 (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 2 (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 3 (For a finite set $S$, the notation $|S|$). For a finite set $S$, the notation $|S|$ denotes the cardinality of $S$.

Definition 4 (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 $C \cdot |A|^{2-\varepsilon}$.

3What 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 $\{a \cdot b : a,b \in A\}$.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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 $\{a \cdot b : a,b \in A\}$.^[1]

1Packet records

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Record	Kind	Assessment
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 $\{a \cdot b : a,b \in A\}$.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For erdos problem 52, pins the dated research frontier: 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. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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 $\{a \cdot b : a,b \in A\}$. Open full record →
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 $\{a \cdot b : a,b \in A\}$.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Erdős's sum-product growth problem for finite sets of integers, record erdos-problem-52-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 52: Let $A$ be a finite set of integers. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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 $\{a \cdot b : a,b \in A\}$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

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.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemErdős's sum-product growth problem for finite sets of integers

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Erdős's sum-product growth problem for finite sets of integers.” TheoremDB. P3002. Problem statement; statement text SHA-256 0d220fd2b72685eb972ac2c0a064fc13e2d9dd5a94344c34e6b595bee2681ada. https://theoremdb.org/statement/?ref=P3002

BibTeX

@misc{theoremdb-problem-0d220fd2b72685eb972ac2c0a064fc13e2d9dd5a94344c34e6b595bee2681ada,
  title = {{Erdős's sum-product growth problem for finite sets of integers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 0d220fd2b72685eb972ac2c0a064fc13e2d9dd5a94344c34e6b595bee2681ada},
  url = {https://theoremdb.org/statement/?ref=P3002}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 52, maintained status record. Erdős Problems record 52, checked 2026-08-01. Problem 52; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 52 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source 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.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 52. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/52.lean:L34; theorem erdos_52; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source 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.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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

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