Representation of Real Numbers by Perron Series, Their Geometry, and Some Applications

The representations of real numbers from (0; 1] by the Perron series (P-representations of numbers) are considered. We establish sufficient conditions for the existence and uniqueness of the expansion of a number in Perron series, clarify the geometric interpretation of digits in the \(\overline{P }\) $\overline{P}$ -representations of numbers in the standard and difference forms, describe the topological-metric properties of cylindrical sets, and analyze the differential properties of continuous monotonic functions (projectors of a \(\overline{P }\) $\overline{P}$ -representation into another \(\overline{P }\) $\overline{P}$ -representation). Sufficient conditions for the singularity of these functions are obtained.