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-Sierpiński-Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth 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.