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Updated Oct 7

Problem 495

Statement. Let $\alpha,\beta \in \mathbb{R}$ . Is it true that

\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0

where $\|x\|$ is the distance from $x$ to the nearest integer?

Status. Open.

Source. erdosproblems.com/495, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #495, https://www.erdosproblems.com/495.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (9)

Irrationality and Diophantine Approximation irrationality/einsiedler_2006_invariant_measures_set_exceptions_littlewood_s Conjecture 1.2 (p. 515): Littlewood's conjecture, liminf n<nu><nv> = 0 for every real u, v Corollary 1.4 (p. 515): such measures are not compactly supported, and are Haar for k prime Proposition 11.1 (p. 557): Littlewood's limit vanishes for (u,v) exactly when an A+-orbit in SL(3,R)/SL(3,Z) is unbounded Theorem 10.1 (p. 555): points with bounded orbits under an open cone of A meet each unstable manifold in dimension zero Theorem 1.3 (p. 515): positive-entropy A-ergodic measures on SL(k,R)/SL(k,Z), k >= 3, are algebraic Theorem 1.5 (p. 516): the exceptional set of Littlewood's conjecture has Hausdorff dimension zero Theorem 2.1 (pp. 523-524): trichotomy for A-ergodic measures on SL(k,R)/Gamma along each pair of root directions

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