Effective bounds for the number of totally real parameters producing preperiodicity

Develop effective, computable, or uniform upper bounds for the cardinality of Prep(a) ∩ Q_tr, where Prep(a)={c∈C: a is preperiodic for f_c(x)=x^2+c}, in the regime a ∈ (−2,2) ∩ Q. Current arguments via arithmetic equidistribution establish finiteness but offer no quantitative control on the number of such totally real parameters c.

Background

Using arithmetic equidistribution, the paper gives an alternative argument for the finiteness of Prep(a)∩Q_tr when a∈(−2,2)∩Q, but this method is ineffective and yields no explicit bounds on the number of such parameters.

The authors suggest that quantitative equidistribution results (e.g., work by DeMarco–Krieger–Ye and Fili building on Favre–Rivera-Letelier) might be leveraged to provide effective or uniform bounds, but this remains to be done.