Littlewood conjecture

Product of three linear forms

Define the cubic form $${\displaystyle L_{\alpha ,\beta }(x,y,z)=x(\alpha x-y)(\beta x-z).}$$ Since ${\displaystyle \|n\alpha \|}$ and ${\displaystyle \|n\beta \|}$ are obtained by choosing the nearest integers ${\displaystyle y}$ and ${\displaystyle z}$, Littlewood's conjecture is equivalent to $${\displaystyle \inf _{(x,y,z)\in \mathbb {Z} ^{3},\,x\neq 0}|L_{\alpha ,\beta }(x,y,z)|=0.}$$ This formulation places the problem in the geometry of numbers and in the study of products of linear forms.^[6]

Diagonal actions on lattices

The dynamical formulation uses the space $${\displaystyle X_{3}=\operatorname {SL} (3,\mathbb {R} )/\operatorname {SL} (3,\mathbb {Z} )}$$ of unimodular lattices in ${\displaystyle \mathbb {R} ^{3}}$. Associate to ${\displaystyle (\alpha ,\beta )}$ the lattice $${\displaystyle x_{\alpha ,\beta }={\begin{pmatrix}1&0&0\\\alpha &1&0\\\beta &0&1\end{pmatrix}}\mathbb {Z} ^{3}.}$$ The integer vector ${\displaystyle (n,-p,-q)}$ becomes $${\displaystyle (n,\;n\alpha -p,\;n\beta -q).}$$ The product of these three coordinates is invariant under determinant-one diagonal transformations. By Mahler's compactness theorem, failure of Littlewood's conjecture corresponds to relative compactness of the orbit of ${\displaystyle x_{\alpha ,\beta }}$ under an appropriate positive semigroup in the diagonal group. This connection permits the use of ergodic theory and homogeneous dynamics.^[1]

Cassels–Swinnerton-Dyer conjecture

Work of Cassels and Swinnerton-Dyer led to a broader conjecture about products of linear forms. Let $${\displaystyle L(\mathbf {x} )=L_{1}(\mathbf {x} )L_{2}(\mathbf {x} )\cdots L_{d}(\mathbf {x} )}$$ be a product of ${\displaystyle d\geq 3}$ linearly independent real linear forms in ${\displaystyle d}$ variables. If ${\displaystyle L}$ is not a nonzero constant multiple of a form with integer coefficients, the conjecture asserts that $${\displaystyle \inf _{\mathbf {x} \in \mathbb {Z} ^{d}\setminus \{\mathbf {0} \}}|L(\mathbf {x} )|=0.}$$ Cassels and Swinnerton-Dyer showed that the case ${\displaystyle d=3}$ of this conjecture would imply Littlewood's conjecture.^[6][7]

Compact diagonal orbits

Let ${\displaystyle D_{d}}$ be the group of positive diagonal matrices of determinant one acting on $${\displaystyle X_{d}=\operatorname {SL} (d,\mathbb {R} )/\operatorname {SL} (d,\mathbb {Z} ).}$$ The compact-orbit conjecture states that every relatively compact ${\displaystyle D_{d}}$-orbit in ${\displaystyle X_{d}}$ is closed. This is the standard dynamical reformulation of the Cassels–Swinnerton-Dyer conjecture; the case ${\displaystyle d=3}$ would imply Littlewood's conjecture.^[8][7]

Disproved broader orbit-closure conjecture

Margulis also proposed a substantially broader description of orbit closures for actions of connected subgroups generated by real-split elements. In that formulation, an orbit closure was expected either to be homogeneous or to arise through a factor on which the acting group becomes a one-parameter non-unipotent group.

This general conjecture is false. In 2010, François Maucourant constructed, for ${\displaystyle G=\operatorname {SL} (d,\mathbb {R} )}$ with ${\displaystyle d\geq 6}$, lattices ${\displaystyle \Gamma \subset G}$, subgroups ${\displaystyle A}$ of the diagonal group, and points ${\displaystyle x\in G/\Gamma }$ for which ${\displaystyle {\overline {Ax}}}$ is not homogeneous and the action does not factor through a one-parameter non-unipotent group.^[9] These counterexamples concern subgroups of diagonal groups in dimensions at least six. They do not disprove the compact-orbit conjecture for the full diagonal group in dimension three and therefore do not settle Littlewood's conjecture.

Metric and dimension results

Gallagher proved that, for almost every pair ${\displaystyle (\alpha ,\beta )}$, $${\displaystyle \liminf _{n\to \infty }n(\log n)^{2}\|n\alpha \|\|n\beta \|=0.}$$ This strengthens Littlewood's conjecture by two logarithmic factors for a full-measure set of pairs.^[4]

Einsiedler, Katok and Lindenstrauss proved that the exceptional set $${\displaystyle {\mathcal {E}}=\left\{(\alpha ,\beta )\in \mathbb {R} ^{2}:\liminf _{n\to \infty }n\|n\alpha \|\|n\beta \|>0\right\}}$$ has Hausdorff dimension zero.^[5] Their proof combines entropy and measure rigidity for higher-rank diagonal actions. It does not show that ${\displaystyle {\mathcal {E}}}$ is empty.

Metric results have also been proved on lower-dimensional subsets. In 2024, Sam Chow and Lei Yang established a two-logarithm strengthening for almost every point on any line in the plane, using effective equidistribution of one-parameter unipotent orbits in ${\displaystyle X_{3}}$.^[10]

Algebraic and explicit pairs

Cassels and Swinnerton-Dyer proved the conjecture when ${\displaystyle 1,\alpha ,\beta }$ span a totally real cubic number field.^[6] Uri Shapira later proved a stronger, fully inhomogeneous statement for the same class of pairs.^[11]

Several results construct large or explicit families inside ${\displaystyle \operatorname {Bad} \times \operatorname {Bad} }$, where ${\displaystyle \operatorname {Bad} }$ denotes the badly approximable numbers. For every fixed ${\displaystyle \alpha \in \operatorname {Bad} }$, Andrew Pollington and Sanju Velani constructed a subset of ${\displaystyle \beta \in \operatorname {Bad} }$ of Hausdorff dimension 1 such that $${\displaystyle n\|n\alpha \|\|n\beta \|\leq {\frac {1}{\log n}}}$$ for infinitely many ${\displaystyle n}$.^[12]

Bernard de Mathan gave effective constructions of linearly independent pairs with bounded partial quotients satisfying the conjecture.^[13] Boris Adamczewski and Yann Bugeaud proved that, for every fixed real ${\displaystyle \alpha }$ with bounded partial quotients, one can explicitly construct continuum many ${\displaystyle \beta }$ with bounded partial quotients for which a strong form of Littlewood's conjecture holds.^[14]

Limits of quantitative strengthening

Gallagher's theorem is an almost-everywhere result and cannot be promoted to a comparable assertion for every pair. Bugeaud and Nikolay Moshchevitin proved that the set of pairs satisfying $${\displaystyle \liminf _{n\to \infty }n(\log n)^{2}\|n\alpha \|\|n\beta \|>0}$$ has full Hausdorff dimension in ${\displaystyle \mathbb {R} ^{2}}$.^[15] Dzmitry Badziahin strengthened this by showing that the set of pairs for which $${\displaystyle \liminf _{n\to \infty }n\log n\log \log n\,\|n\alpha \|\|n\beta \|>0}$$ has full Hausdorff dimension.^[16] These full-dimensional sets have Lebesgue measure zero, so the results are compatible with Gallagher's theorem.

In 2026, Reynold Fregoli and Dmitry Kleinbock generalized Badziahin's logarithmic obstruction to vectors in ${\displaystyle \mathbb {R} ^{d}}$, replacing ${\displaystyle \log n\log \log n}$ by ${\displaystyle (\log n)^{d-1}\log \log n}$.^[17]

The ${\displaystyle p}$-adic Littlewood conjecture

In 2004, de Mathan and Olivier Teulié proposed a mixed real and ${\displaystyle p}$-adic analogue. For a prime ${\displaystyle p}$, it asks whether $${\displaystyle \inf _{n\geq 1}n\,|n|_{p}\,\|n\alpha \|=0}$$ for every real number ${\displaystyle \alpha }$, where ${\displaystyle |n|_{p}=p^{-v_{p}(n)}}$ is the ${\displaystyle p}$-adic absolute value. They proved the assertion for quadratic irrational numbers.^[23] The conjecture remains open in general. Einsiedler and Dmitry Kleinbock proved that exceptional sets for several ${\displaystyle p}$-adic Littlewood-type problems have Hausdorff dimension zero.^[24]

Function-field analogues

Analogues over fields of formal Laurent series behave differently from the classical real conjecture. Faustin Adiceam, Erez Nesharim and Fred Lunnon constructed an explicit counterexample to the ${\displaystyle t}$-adic analogue over finite fields of characteristic 3.^[25] Samuel Garrett and Steven Robertson later produced counterexamples to the ${\displaystyle P(t)}$-adic analogue in characteristic 5 and proved that the characteristic-3 construction also works in characteristics 7 and 11.^[26]

Robertson subsequently proved a transference principle from the ${\displaystyle t}$-adic problem to analogues associated with arbitrary irreducible polynomials ${\displaystyle P(t)}$ and developed a general framework using number walls, infinite arrays of Toeplitz determinants that translate approximation questions into combinatorial ones.^[2] These counterexamples do not affect the classical conjecture over the real numbers.