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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: $\forall u,v\in ℝ$,

$\underset{n\to \infty }{lim inf}n〈nu〉〈nv〉=0,$

where $〈w〉={min}_{n\in ℤ}|w-n|$ is the distance of $w\in ℝ$ to the nearest integer. Let $A$ be the group of positive diagonal $k×k$ matrices on $\text{SL}\left(k,ℝ\right)/\text{SL}\left(k,ℤ\right)$. 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 $X=\text{SL}\left(k,ℝ\right)/\text{SL}\left(k,ℤ\right)$ for a subgroup of $A$ which acts on $X$ with positive entropy. Then $\mu$ is algebraic.

2) Let ${\Xi }=\left\{\left(u,v\right)\in {ℝ}^{2}:{lim inf}_{n\to \infty }n〈nu〉〈nv〉>0\right\}$. Then the Hausdorff dimension

${dim}_{H}{\Xi }=0·$

3) For any $k$ linear forms ${m}_{i}\left({x}_{1},\cdots ,{x}_{k}\right)={\sum }_{j=1}^{k}{m}_{ij}\left({x}_{j}\right)$ and ${f}_{m}\left({x}_{1},\cdots ,{x}_{k}\right)={\prod }_{i=1}^{k}{m}_{i}\left({x}_{1},\cdots ,{x}_{k}\right)$, where $m=\left({m}_{ij}\right)$ denotes the $k×k$ matrix whose rows are the linear forms ${m}_{i}\left({x}_{1},\cdots ,{x}_{k}\right)$, there is a set ${{\Xi }}_{k}\subset \text{SL}\left(k,ℝ\right)$ of Hausdorff dimension $k-1$ so that $\forall m\in \text{SL}\left(k,ℝ\right)\setminus {{\Xi }}_{k}$,

$\underset{x\in {ℤ}^{k}\setminus \left\{0\right\}}{inf}|{f}_{m}\left(x\right)|=0·$

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 28A80 Fractals 37A15 General groups of measure-preserving transformation 11H46 Products of linear forms