Representations of Real Numbers by Alternating Perron Series and Their Geometry

We consider the representation of real numbers by alternating Perron series ($P^-$-representation), which is a generalization of representations of real numbers by Ostrogradsky-Sierpi\'nski-Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating L\"{u}roth series, etc. Namely, we prove the basic topological and metric properties of $P^-$-representation and find the relationship between $P$-representation and $P^-$-representation in some measure theory problems.