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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: $$\forall u,v\in\mathbb R$$, $\liminf_{n\to\infty} n \langle nu \rangle \langle nv \rangle =0,$ where $$\langle w\rangle =\min_{n\in\mathbb Z} | w-n|$$ is the distance of $$w\in\mathbb R$$ to the nearest integer. Let $$A$$ be the group of positive diagonal $$k\times k$$ matrices on $$\text{SL}(k,\mathbb R)/\text{SL}(k,\mathbb Z)$$. 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 $$X=\text{SL}(k,\mathbb R)/\text{SL}(k,\mathbb Z)$$ for a subgroup of $$A$$ which acts on $$X$$ with positive entropy. Then $$\mu$$ is algebraic.
2) Let $$\Xi = \{(u,v)\in\mathbb R^2: \liminf_{n\to\infty} n \langle nu\rangle \langle nv \rangle >0\}$$. 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 $$\Xi_k \subset\text{SL}(k,\mathbb R)$$ of Hausdorff dimension $$k-1$$ so that $$\forall m\in\text{SL}(k,\mathbb R)\setminus \Xi_k$$, $\inf_{x\in {\mathbb Z}^k \setminus \{ 0 \} } | f_m (x)| =0.$
The last result has applications to a generalization of Littlewood’s conjecture.

##### MSC:
 22F10 Measurable group actions 11J13 Simultaneous homogeneous approximation, linear forms 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 28A80 Fractals 37A15 General groups of measure-preserving transformations and dynamical systems 11H46 Products of linear forms
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