[#P3054] Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets

Problem. Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?

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1Context

This problem originates from Erdős problem 153 on erdosproblems.com and concerns the structure of sumsets of Sidon sets. The question asks whether the minimum mean square gap in the ordered sumset $A+A$ must grow without bound as the Sidon set $A$ becomes arbitrarily large.

2Definitions

Definition 1 (A Sidon set). A Sidon set is a set $A \subseteq \mathbb{N}$ such that all sums $a+b$ with $a,b \in A$ and $a \leq b$ are distinct; equivalently, the equation $a+b=c+d$ with $a,b,c,d \in A$ and $a \leq b$, $c \leq d$ implies $\{a,b\} = \{c,d\}$.

Definition 2 (For a finite set $S \subset \mathbb{N}$ with $S = \{s_1 < s_2 < \cdots < s_t\}$, the mean square gap). For a finite set $S \subset \mathbb{N}$ with $S = \{s_1 < s_2 < \cdots < s_t\}$, the mean square gap is defined as $\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2$, which measures the average squared distance between consecutive elements of $S$.

Definition 3 (The sumset of a set $A$ with itself). The sumset of a set $A$ with itself is $A+A = \{a+b : a,b \in A\}$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?

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 153 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 153 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?	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 153 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?^[1] Relevance to this problem For erdos problem 153, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 153 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: 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 153 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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?	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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$?^[1] Relevance to this problem For Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets, record erdos-problem-153-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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. 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 153: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a+b$ with $a \leq b$ in $A$ are distinct. For a finite set $S = \{s_1 < s_2 < \cdots < s_t\}$ of natural numbers, define the mean square gap to be \[\frac{1}{t}\sum_{i=1}^{t-1}(s_{i+1}-s_i)^2.\] For each positive integer $n$, let $f(n)$ denote the minimum mean square gap of $A+A$, as $A$ ranges over all Sidon sets of cardinality $n$, where $A+A = \{a+b : a,b \in A\} = \{s_1 < s_2 < \cdots < s_t\}$ is the sumset. Does $f(n) \to \infty$ as $n \to \infty$? 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 153 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 153 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.

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

ProblemAsymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets

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Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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“Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets.” TheoremDB. P3054. Problem statement; statement text SHA-256 c58e1778de50adaf6524b793aae311f7822e029435ae4ea26425424391fc0761. https://theoremdb.org/statement/?ref=P3054

BibTeX

@misc{theoremdb-problem-c58e1778de50adaf6524b793aae311f7822e029435ae4ea26425424391fc0761,
  title = {{Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c58e1778de50adaf6524b793aae311f7822e029435ae4ea26425424391fc0761},
  url = {https://theoremdb.org/statement/?ref=P3054}
}

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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.

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1References

Packet source. Erdős Problems database, Problem 153, maintained status record. Erdős Problems record 153, checked 2026-08-01. Problem 153; 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 153; 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 Asymptotic Growth of Mean Square Gaps in Sumsets of 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 153 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 Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 153. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/153.lean:L45; theorem erdos_153; 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 Asymptotic Growth of Mean Square Gaps in Sumsets of 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.

[#P3054] Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets

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