[#P2992] Erdős Problem 40 on additive bases with slow growth

A finite mathematical diagram showing an additive set and the multiplicities in its sumset. — An additive set mapped to sumset multiplicities.

Problem. Erdős Problem 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $|A \cap \{1, \ldots, N\}| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$.

1Context

This problem originates from Erdős and concerns additive bases. The Erdős–Turán conjecture on additive bases (Erdős Problem 28) asserts that if $A$ is an asymptotic basis of order 2 (meaning every sufficiently large integer is a sum of two elements of $A$), then the representation function is unbounded. Problem 40 asks for a refinement: how slowly can the counting function $|A \cap \{1, \ldots, N\}|$ grow, relative to $\sqrt{N}$, while still guaranteeing that the representation function has unbounded limsup?

2Problem setup

Definition 1 (For a set $A \subseteq \mathbb{N}$ and $n \in \mathbb{N}$, the representation function $(1_A * 1_A)(n)$). For a set $A \subseteq \mathbb{N}$ and $n \in \mathbb{N}$, the representation function $(1_A * 1_A)(n)$ is defined as $\sum_{a + b = n} 1_A(a) \cdot 1_A(b)$, which counts ordered pairs $(a, b) \in A \times A$ with $a + b = n$.

Definition 2 (For functions $f, h: \mathbb{N} \to \mathbb{R}$, we write $f(N) =O h(N)$ as $N \to \infty$ if there exists a positive constant $C$ and $N_0 \in \mathbb{N}$ such that $|f(N)| \leq C \cdot |h(N)|$ for all $N \geq N_0$). For functions $f, h: \mathbb{N} \to \mathbb{R}$, we write $f(N) =O h(N)$ as $N \to \infty$ if there exists a positive constant $C$ and $N_0 \in \mathbb{N}$ such that $|f(N)| \leq C \cdot |h(N)|$ for all $N \geq N_0$.

Definition 3 (For a set $A \subseteq \mathbb{N}$ and a function $g: \mathbb{N} \to \mathbb{R}$, the condition $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$). For a set $A \subseteq \mathbb{N}$ and a function $g: \mathbb{N} \to \mathbb{R}$, the condition $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$ means that the function $N \mapsto \frac{\sqrt{N}}{g(N)}$ is $O$-big-oh of the function $N \mapsto |A \cap \{1, \ldots, N\}|$ as $N \to \infty$.

Remark 1. This problem originates from Erdős and concerns additive bases. The Erdős–Turán conjecture on additive bases (Erdős Problem 28) asserts that if $A$ is an asymptotic basis of order 2 (meaning every sufficiently large integer is a sum of two elements of $A$), then the representation function is unbounded. Problem 40 asks for a refinement: how slowly can the counting function $|A \cap \{1, \ldots, N\}|$ grow, relative to $\sqrt{N}$, while still guaranteeing that the representation function has unbounded limsup?

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $|A \cap \{1, \ldots, N\}| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$.

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 40 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $|A \cap \{1, \ldots, N\}| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $|A \cap \{1, \ldots, N\}| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$.^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 40 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $\|A \cap \{1, \ldots, N\}\|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $\|A \cap \{1, \ldots, N\}\| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$.^[1] Relevance to this problem For erdos problem 40, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 40 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For. 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 40 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 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $\|A \cap \{1, \ldots, N\}\|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $\|A \cap \{1, \ldots, N\}\| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$. Open full record →
	Trace	Reported
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $\|A \cap \{1, \ldots, N\}\|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $\|A \cap \{1, \ldots, N\}\| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$.^[1] Relevance to this problem For Erdős Problem 40 on additive bases with slow growth, record erdos-problem-40-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 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $\|A \cap \{1, \ldots, N\}\|$ denote the cardinality of the intersection. 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 40: For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $\|A \cap \{1, \ldots, N\}\|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property holds for sets $A$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$ if there exists a positive constant $C$ such that $\|A \cap \{1, \ldots, N\}\| \geq C \cdot \frac{\sqrt{N}}{g(N)}$ for all sufficiently large $N$. A function $g: \mathbb{N} \to \mathbb{R}$ tends to infinity if for every $M > 0$, there exists $N_0$ such that $g(N) > M$ for all $N \geq N_0$. Determine the set of all functions $g: \mathbb{N} \to \mathbb{R}$ tending to infinity such that for every set $A \subseteq \mathbb{N}$ satisfying $\|A \cap \{1, \ldots, N\}\| \gg \frac{\sqrt{N}}{g(N)}$, we have $\limsup_{n \to \infty} (1_A * 1_A)(n) = \infty$. 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 materialContext, examples, and computations

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 40 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 40 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 connectTyped relations and evidence flow

ProblemErdős Problem 40 on additive bases with slow growth

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Erdős Problem 40 on additive bases with slow growth,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-40

This page as plain text: erdos-problem-40.md

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

1References

Packet source. Erdős Problems database, Problem 40, maintained status record. Erdős Problems record 40, checked 2026-08-01. Problem 40; 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 40; 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 Problem 40 on additive bases with slow growth: 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 40 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 Problem 40 on additive bases with slow growth: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 40. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/40.lean:L55; theorem erdos_40; 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 Problem 40 on additive bases with slow growth: 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.

[#P2992] Erdős Problem 40 on additive bases with slow growth

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance