Remarks on Garsia entropy and multidimensional Erdös measures

We conjecture that the Garsia entropy coincides with the entropy of the invariant multidimensional Erdös measure. This conjecture is true for all Garsia numbers. We also specify the algebraic unit being non-Pisot number, for which this conjecture is true.
We prove a theorem, which generalizes the Garsia theorem on the absolute continuity of the infinite Bernoulli convolution for the Garsia numbers. The proof uses relations between the multidimensional Erdös problem and the one-dimensional Erdös problem.
We discuss a connection between the entropy of the invariant Erdös measure and the conditional Ledrappier–Young entropies. We also formulate three conjectures and obtain some consequences from them. In particular, we conjecture that the Hausdorff dimension of the Erdös measure is equal to the Ledrappier–Young dimension of conditional measure for the multidimensional invariant Erdös measure along the unstable foliation corresponding to the top Lyapunov exponent of multiplicity 1. For 2-numbers, we obtain formulae for the Hausdorff dimension of Erdös measures on the unstable plane.