[#P2980] Unbounded representation function for sets with cofinite sumset

Problem. Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

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Definitions and notation

1Context

This problem originates from Erdős's collection of open problems in additive combinatorics. It concerns the behavior of representation functions for sets with cofinite sumsets.

2Problem setup

Definition 1 (The sumset $A + A$ of a set $A \subseteq \mathbb{N}$). The sumset $A + A$ of a set $A \subseteq \mathbb{N}$ is the set $\{a + b : a, b \in A\}$ of all sums of two (not necessarily distinct) elements of $A$.

Remark 1. The representation function $r_A(n)$ counts the number of ordered pairs $(a, b) \in A \times A$ with $a + b = n$.

Definition 2 (A set $S \subseteq \mathbb{N}$). A set $S \subseteq \mathbb{N}$ is cofinite if its complement $S^c = \mathbb{N} \setminus S$ is finite.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

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 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.	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 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.^[1] Relevance to this problem For erdos problem 28, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: 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 28 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 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.	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 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.^[1] Relevance to this problem For Unbounded representation function for sets with cofinite sumset, record erdos-problem-28-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 28: Let $A$ be a set of natural numbers. 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 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$. 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 28 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 28 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.

ProblemUnbounded representation function for sets with cofinite sumset

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

“Unbounded representation function for sets with cofinite sumset.” TheoremDB. P2980. Problem statement; statement text SHA-256 687d21a4ba1087803da6e1a0c310d28ea24dea658d4b7ff77b016cb73600f8a4. https://theoremdb.org/statement/?ref=P2980

BibTeX

@misc{theoremdb-problem-687d21a4ba1087803da6e1a0c310d28ea24dea658d4b7ff77b016cb73600f8a4,
  title = {{Unbounded representation function for sets with cofinite sumset}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 687d21a4ba1087803da6e1a0c310d28ea24dea658d4b7ff77b016cb73600f8a4},
  url = {https://theoremdb.org/statement/?ref=P2980}
}

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.

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1References

Packet source. Erdős Problems database, Problem 28, maintained status record. Erdős Problems record 28, checked 2026-08-01. Problem 28; 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 28; 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 Unbounded representation function for sets with cofinite sumset: 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 28 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 Unbounded representation function for sets with cofinite sumset: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 28. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/28.lean:L36; theorem erdos_28; 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 Unbounded representation function for sets with cofinite sumset: 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.

[#P2980] Unbounded representation function for sets with cofinite sumset

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