There is a well-known and long-standing conjecture of Littlewood: ,
where is the distance of to the nearest integer. Let be the group of positive diagonal matrices on . In the paper under review some results which have implications on Littlewood’s conjecture are proven. Main results of the paper are:
1) Let be an -invariant and ergodic measure on for a subgroup of which acts on with positive entropy. Then is algebraic.
2) Let . Then the Hausdorff dimension
3) For any linear forms and , where denotes the matrix whose rows are the linear forms , there is a set of Hausdorff dimension so that ,
The last result has applications to a generalization of Littlewood’s conjecture.