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

- ID: `P3054`
- Reference: `erdos-problem-153`
- Page: https://theoremdb.org/statements/P3054
- Record maturity: Reviewed problem with recorded work

## 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 $n \to \infty$?

### Context

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.

### Problem setup

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

### What 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 $n \to \infty$?

## 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. 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](#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 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 $n \to \infty$?

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 $n \to \infty$?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-153`, 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 153, maintained status record. Erdős Problems record 153, checked 2026-08-01. Problem 153; status field and linked bibliography https://www.erdosproblems.com/153
   - 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
   - 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 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 153 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 Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets: 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 153. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/153.lean:L45; theorem erdos_153; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/153.lean#L45
   - 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 Asymptotic Growth of Mean Square Gaps in Sumsets of Sidon Sets: Supplies the pinned formal declaration whose human-readable editorial statement is published here.