Probabilistic FUP for discrete random Cantor sets in Ensembles II and III

Develop a probabilistic approach to establish the discrete fractal uncertainty principle estimate (1.5) for random discrete Cantor sets constructed in Ensemble II (independent random alphabets at each iteration) and Ensemble III (independent random alphabets assigned to each parent interval), addressing the lack of submultiplicativity by analyzing higher iterations.

Background

In the discrete setting on Z_N, Ensemble I admits a submultiplicativity property that allows one to deduce FUP bounds at all scales from the first iteration, and a probabilistic proof of FUP with improved exponent has been given in [EH].

For Ensembles II and III, different alphabets can be used at different iterations or branches, breaking submultiplicativity and preventing the reduction to the first iteration. As a result, proving FUP requires controlling higher iterations in a genuinely probabilistic framework, which the authors highlight as open.