[#P11] Littlewood conjecture
Contents
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.
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Definitions and notation
1Context
The problem asks whether two real numbers always admit sufficiently strong simultaneous rational approximations with a shared denominator.
2Problem setup
Definition 1 (The notation ||x||). The notation ||x|| means min{|x-m| : m is an integer}.
Remark 1. The lower limit is taken over positive integers n.
3What 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.
1Status
Saved packet · July 31, 2026
Saved packet status (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.[1]
1Packet records
Recent contributions
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
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 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- 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 1. If alpha is rational, ||n alpha|| is zero for infinitely many n, so the conclusion follows for every beta.
Computational notes
- Finite calculations can produce small products for a chosen pair without determining its infinite lower limit.
2See also
- Closest prime square to the cube of a prime below one trilliondiophantine approximation
- Unbounded continued-fraction coefficients of pidiophantine approximation
- Lonely runner conjecturediophantine approximation
Contribute to this problem
Cite this problem statement
Cite the original sources separately.
“Littlewood conjecture.” TheoremDB. P11. Problem statement; statement text SHA-256 94ac102b45aef3269c1cc91403b0e83550c79f2952c6bc04202a0d568888a149. https://theoremdb.org/statement/?ref=P11
@misc{theoremdb-problem-94ac102b45aef3269c1cc91403b0e83550c79f2952c6bc04202a0d568888a149,
title = {{Littlewood conjecture}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 94ac102b45aef3269c1cc91403b0e83550c79f2952c6bc04202a0d568888a149},
url = {https://theoremdb.org/statement/?ref=P11}
}Plain text: Built Markdown snapshot
This problem includes 2 records joined by 2 typed links, sourced from annals.math.princeton.edu[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗website · primary source · checked 2026-07-31Source 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.Also cited at Abstract and main theorem.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.Proves the Hausdorff-dimension-zero exceptional-set theorem.Source named by the research packet.
An original CC0 restatement prepared by TheoremDB maintainers.
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