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.


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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