The Littlewood conjecture

For any two real numbers $\alpha$ and $\beta$, Dirichlet's approximation theorem already shows that $n\lVert n\alpha\rVert\lVert n\beta\rVert<1/2$ for infinitely many $n$. Littlewood conjectured around 1930 that this product can be made arbitrarily small. The question links Diophantine approximation with the dynamics of flows on spaces of lattices.

Problem

For a real number $x$ let $\lVert x\rVert$ denote its distance to the nearest integer. Is it true that for all real numbers $\alpha$ and $\beta$

$$\liminf_{n\to\infty}\; n\,\lVert n\alpha\rVert\,\lVert n\beta\rVert=0\,?$$

Background

The conjecture is due to J. E. Littlewood and dates from around 1930; Erdős recorded it in his 1961 list of problems. It is easy when $\alpha$ or $\beta$ is not badly approximable: if $\liminf n\lVert n\alpha\rVert=0$, the bound $\lVert n\beta\rVert\le 1/2$ finishes the job. The difficulty lies with pairs of badly approximable numbers, such as two quadratic irrationals. The strongest results so far come from studying measures on spaces of lattices that are invariant under diagonal flows, an approach developed by Einsiedler, Katok and Lindenstrauss.

What is known

• Cubic fields. Cassels and Swinnerton-Dyer (1955) proved the conjecture when $1,\alpha,\beta$ span a totally real cubic number field.
• Almost all pairs. Gallagher (1962) proved a stronger statement for almost every pair $(\alpha,\beta)$ in the sense of Lebesgue measure.
• Large sets of good pairs. Pollington and Velani (2000) showed that for every badly approximable $\alpha$ there is a set of $\beta$ of Hausdorff dimension 1 with $n\lVert n\alpha\rVert\lVert n\beta\rVert\le 1/\log n$ for infinitely many $n$.
• Small exceptional set. Einsiedler, Katok and Lindenstrauss (2006) proved, using measure rigidity, that the set of pairs $(\alpha,\beta)$ for which the conjecture fails has Hausdorff dimension zero. Venkatesh (2008) gives an exposition.

What would count as progress

• A proof of the conjecture for a new explicit class of pairs, for example for some pair of quadratic irrationals from different quadratic fields.
• A counterexample: an explicit pair $(\alpha,\beta)$ with $n\lVert n\alpha\rVert\lVert n\beta\rVert\ge c>0$ for all $n\ge1$, given with a proof.
• Numerical studies of $\min_{n\le N} n\lVert n\alpha\rVert\lVert n\beta\rVert$ for specific pairs, with code, as evidence for or against particular approaches.
• A formal proof against the Lean statement in formal-conjectures.

References

1. J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London A 248 (1955), 73–96. doi:10.1098/rsta.1955.0010
2. P. Gallagher, Metric simultaneous diophantine approximation, J. London Math. Soc. 37 (1962), 387–390. doi:10.1112/jlms/s1-37.1.387
3. A. D. Pollington and S. L. Velani, On a problem in simultaneous diophantine approximation: Littlewood's conjecture, Acta Math. 185 (2000), 287–306. doi:10.1007/BF02392812
4. M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. 164 (2006), 513–560. doi:10.4007/annals.2006.164.513
5. A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. 45 (2008), 117–134. doi:10.1090/S0273-0979-07-01194-9

Source

Imported from Erdős Problems 495, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.