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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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