[#P2990] Existence of dense infinite Sidon sets

Problem. Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ |A \cap \{1, 2, \ldots, N\}| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.

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Definitions and notation

1Context

This is a classical problem in additive combinatorics concerning the maximum possible density of infinite Sidon sets. Sidon sets are also known as $B_2$ sets in the literature. The problem asks whether the trivial upper bound of $O(N^{1/2})$ for the size of a Sidon set in $\{1, \ldots, N\}$ can be nearly attained by an infinite set, up to an arbitrarily small polynomial factor $N^{-\varepsilon}$.

2Definitions

Definition 1 (A Sidon set). A Sidon set is a set of natural numbers in which all pairwise sums of (not necessarily distinct) elements are distinct, except for the trivial reordering of terms.

Definition 2 (For functions $f, g: \mathbb{N} \to \mathbb{R}_{\geq 0}$, the notation $f(N) \gg_\varepsilon g(N)$). For functions $f, g: \mathbb{N} \to \mathbb{R}_{\geq 0}$, the notation $f(N) \gg_\varepsilon g(N)$ means that there exists a constant $C_\varepsilon > 0$ depending on $\varepsilon$ such that $g(N) \leq C_\varepsilon \cdot f(N)$ for all sufficiently large $N$.

Definition 3 (The notation $g(N) = O(f(N))$ as $N \to \infty$). The notation $g(N) = O(f(N))$ as $N \to \infty$ means that there exists a constant $C > 0$ and $N_0 \in \mathbb{N}$ such that $|g(N)| \leq C \cdot |f(N)|$ for all $N \geq N_0$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ |A \cap \{1, 2, \ldots, N\}| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.

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 39 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ |A \cap \{1, 2, \ldots, N\}| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 39 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.	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 39 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.^[1] Relevance to this problem For erdos problem 39, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 39 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A. 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 39 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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.	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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$.^[1] Relevance to this problem For Existence of dense infinite Sidon sets, record erdos-problem-39-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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. 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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function satisfies \[ \|A \cap \{1, 2, \ldots, N\}\| \gg_\varepsilon N^{1/2 - \varepsilon} \] as $N \to \infty$? Here, the notation $f(N) \gg_\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\varepsilon$. 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 39 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 39 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.

ProblemExistence of dense infinite Sidon sets

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

“Existence of dense infinite Sidon sets.” TheoremDB. P2990. Problem statement; statement text SHA-256 bb05993c317472987fc9a9afbc6caff00b3ce302713012b4fce7f4792a54022b. https://theoremdb.org/statement/?ref=P2990

BibTeX

@misc{theoremdb-problem-bb05993c317472987fc9a9afbc6caff00b3ce302713012b4fce7f4792a54022b,
  title = {{Existence of dense infinite Sidon sets}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 bb05993c317472987fc9a9afbc6caff00b3ce302713012b4fce7f4792a54022b},
  url = {https://theoremdb.org/statement/?ref=P2990}
}

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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 39, maintained status record. Erdős Problems record 39, checked 2026-08-01. Problem 39; 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 39; 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 Existence of dense infinite Sidon sets: 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 39 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 Existence of dense infinite Sidon sets: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 39. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/39.lean:L35; theorem erdos_39; 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 Existence of dense infinite Sidon sets: 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.

[#P2990] Existence of dense infinite Sidon sets

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