# P3050: Convergence of a weighted sum over well-separated sets

- ID: `P3050`
- Reference: `erdos-problem-143`
- Page: https://theoremdb.org/statements/P3050
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
\[
\sum_{x \in A} \frac{1}{x \log x}
\]
convergent?

### Context

This problem is part of a family of questions about the density and distribution of well-separated sets of real numbers, originally posed by Erdős.

### Problem setup

- **Definition (A subset $A \subseteq \mathbb{R}$).** A subset $A \subseteq \mathbb{R}$ is well-separated if $A \subseteq (1, \infty)$, the set $A$ is countably infinite, and for all distinct $x, y \in A$ and all integers $k \geq 1$, the inequality $|kx - y| \geq 1$ holds.
- **Remark.** This problem is part of a family of questions about the density and distribution of well-separated sets of real numbers, originally posed by Erdős.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
\[
\sum_{x \in A} \frac{1}{x \log x}
\]
convergent?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 143 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
\[
\sum_{x \in A} \frac{1}{x \log x}
\]
convergent? [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 143 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
\[
\sum_{x \in A} \frac{1}{x \log x}
\]
convergent?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 143 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 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
\[
\sum_{x \in A} \frac{1}{x \log x}
\]
convergent?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 143 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 143 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 143: A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series \[ \sum_{x \in A} \frac{1}{x \log x} \] convergent? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-143`, 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 143, maintained status record. Erdős Problems record 143, checked 2026-08-01. Problem 143; status field and linked bibliography https://www.erdosproblems.com/143
   - Also cited at Problem 143; 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 Convergence of a weighted sum over well-separated 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 143 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 Convergence of a weighted sum over well-separated 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 143. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/143.lean:L56; theorem erdos_143.parts.ii; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/143.lean#L56
   - 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 Convergence of a weighted sum over well-separated sets: Supplies the pinned formal declaration whose human-readable editorial statement is published here.