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Invariant measures and the set of exceptions to Littlewood’s conjecture. (English) Zbl 1109.22004
There is a well-known and long-standing conjecture of Littlewood: $\forall u,v\in\Bbb R$, $$\liminf_{n\to\infty} n \langle nu \rangle \langle nv \rangle =0,$$ where $\langle w\rangle =\min_{n\in\Bbb Z} \vert w-n\vert $ is the distance of $w\in\Bbb R$ to the nearest integer. Let $A$ be the group of positive diagonal $k\times k$ matrices on $\text{SL}(k,\Bbb R)/\text{SL}(k,\Bbb Z)$. In the paper under review some results which have implications on Littlewood’s conjecture are proven. Main results of the paper are: 1) Let $\mu$ be an $A$-invariant and ergodic measure on $X=\text{SL}(k,\Bbb R)/\text{SL}(k,\Bbb Z)$ for a subgroup of $A$ which acts on $X$ with positive entropy. Then $\mu$ is algebraic. 2) Let $\Xi = \{(u,v)\in\Bbb R^2: \liminf_{n\to\infty} n \langle nu\rangle \langle nv \rangle >0\}$. Then the Hausdorff dimension $$\dim_H \Xi =0.$$ 3) For any $k$ linear forms $m_i (x_1 ,\dots ,x_k )=\sum_{j=1}^k m_{ij}(x_j )$ and $f_m (x_1 ,\dots ,x_k )=\prod_{i=1}^k m_i (x_1 ,\dots ,x_k )$, where $m=(m_{ij})$ denotes the $k\times k$ matrix whose rows are the linear forms $m_i (x_1 ,\dots ,x_k )$, there is a set $\Xi_k \subset\text{SL}(k,\Bbb R)$ of Hausdorff dimension $k-1$ so that $\forall m\in\text{SL}(k,\Bbb R)\setminus \Xi_k $, $$\inf_{x\in {\Bbb Z}^k \setminus \{ 0 \} } \vert f_m (x)\vert =0.$$ The last result has applications to a generalization of Littlewood’s conjecture.

22F10Measurable group actions
11J13Simultaneous homogeneous approximation, linear forms
37A45Relations of ergodic theory with number theory and harmonic analysis
37A15General groups of measure-preserving transformation
11H46Products of linear forms
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