# P11: Littlewood conjecture

- ID: `P11`
- Reference: `littlewood-conjecture`
- Page: https://theoremdb.org/statements/P11
- Record maturity: Reviewed problem with recorded work

## Problem

For every \(\alpha,\beta\in\mathbb{R}\), one has \(\liminf_{n\to\infty} n\lVert n\alpha\rVert\lVert n\beta\rVert=0\), where \(\lVert x\rVert\) is the distance from \(x\) to the nearest integer.

### Context

The problem asks whether two real numbers always admit sufficiently strong simultaneous rational approximations with a shared denominator.

### Problem setup

- **Definition (The notation ||x||).** The notation ||x|| means min{|x-m| : m is an integer}.
- **Definition (The lower limit).** The lower limit is taken over positive integers n.
- **Remark.** The problem asks whether two real numbers always admit sufficiently strong simultaneous rational approximations with a shared denominator.

### What counts as a solution

- Prove the stated liminf equality for every real pair alpha and beta, or give a pair and prove that the liminf is strictly positive.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The set of pairs failing the classical conjecture has Hausdorff dimension zero. Exact unresolved remainder: For every real alpha and beta, prove liminf as n tends to infinity of n||n alpha||||n beta|| equals 0. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The set of pairs failing the classical conjecture has Hausdorff dimension zero. Exact unresolved remainder: For every real alpha and beta, prove liminf as n tends to infinity of n||n alpha||||n beta|| equals 0.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The set of pairs failing the classical conjecture has Hausdorff dimension zero.

Exact unresolved remainder: For every real alpha and beta, prove liminf as n tends to infinity of n||n alpha||||n beta|| equals 0.

### Background and intake notes

- Original intake status: The cited research paper proves that the exceptional set has Hausdorff dimension zero while leaving the universal conjecture unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and major exceptional-set result were checked against the cited Annals paper on 2026-07-22.
- The conjecture is known for many pairs, including cases with rational dependence. A counterexample would have to lie in a very small exceptional set.

- Recorded example: If alpha is rational, ||n alpha|| is zero for infinitely many n, so the conclusion follows for every beta.

### Open directions

- **Route 1** (reported): Prove the stated liminf equality for every real pair alpha and beta, or give a pair and prove that the liminf is strictly positive. [1](#reference-1)

### Computational notes

- Finite calculations can produce small products for a chosen pair without determining its infinite lower limit.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `littlewood-conjecture`, 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>Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics 164 (2006), 513-560. DOI 10.4007/annals.2006.164.513. Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Annals of Mathematics 164 (2006), abstract and section 1 https://annals.math.princeton.edu/2006/164-2/p04
   - Also cited at Abstract and main theorem
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The cited research paper proves that the exceptional set has Hausdorff dimension zero while leaving the universal conjecture unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Proves the Hausdorff-dimension-zero exceptional-set theorem.
   - Source named by the research packet.