# P3006: Erdős's Additive Representation Density Problem

- ID: `P3006`
- Reference: `erdos-problem-66`
- Page: https://theoremdb.org/statements/P3006
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit
\[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\]
exists?

### Context

This problem concerns the asymptotic behavior of additive representation functions in additive combinatorics. Erdős posed the question of whether a set of positive integers can have its representation function grow like a nonzero constant multiple of the logarithm.

### Problem setup

- **Definition (For a subset $A \subseteq \mathbb{N}$ and $n \in \mathbb{N}$, the additive representation function $r_A(n)$ counts the number of ordered pairs $(a, b) \in A \times A$ with $a + b = n$).** For a subset $A \subseteq \mathbb{N}$ and $n \in \mathbb{N}$, the additive representation function $r_A(n)$ counts the number of ordered pairs $(a, b) \in A \times A$ with $a + b = n$.
- **Definition (The notation $\lim_{n \to \infty} f(n) = c$).** The notation $\lim_{n \to \infty} f(n) = c$ means that for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $|f(n) - c| < \epsilon$ for all $n \geq N$.
- **Remark.** This problem concerns the asymptotic behavior of additive representation functions in additive combinatorics. Erdős posed the question of whether a set of positive integers can have its representation function grow like a nonzero constant multiple of the logarithm.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit
\[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\]
exists?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 66 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit
\[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\]
exists? [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 66 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit
\[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\]
exists?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 66 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 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit
\[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\]
exists?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 66 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 66 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 66: Let $\mathbb{N}$ denote the set of positive integers. For a subset $A \subseteq \mathbb{N}$ and a positive integer $n$, define the additive representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Does there exist a subset $A \subseteq \mathbb{N}$ and a nonzero real number $c$ such that the limit \[\lim_{n \to \infty} \frac{r_A(n)}{\log n} = c\] exists? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-66`, 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 66, maintained status record. Erdős Problems record 66, checked 2026-08-01. Problem 66; status field and linked bibliography https://www.erdosproblems.com/66
   - Also cited at Problem 66; 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 Additive Representation Density Problem: 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 66 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 Additive Representation Density Problem: 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 66. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/66.lean:L37; theorem erdos_66; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/66.lean#L37
   - 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 Additive Representation Density Problem: Supplies the pinned formal declaration whose human-readable editorial statement is published here.