# Limiting curves for polynomial adic systems

## TL;DR

This paper investigates the limiting behavior of partial sum deviations in polynomial adic systems, establishing conditions for the existence of limiting curves and generalizing prior results in ergodic theory.

## Contribution

It provides a general framework for the existence of limiting curves in polynomial adic systems, extending previous work and addressing open questions in the field.

## Key findings

- Established necessary conditions for limiting curve existence.
- Generalized previous results by Janvresse, de la Rue, and Velenik.
- Described limiting curves for ergodic measure-preserving transformations.

## Abstract

We prove the existence and describe limiting curves resulting from deviations in partial sums in the ergodic theorem for cylindrical functions and polynomial (self-similar) adic systems. For a general ergodic measure-preserving transformation and a summable function we give a necessary condition for a limiting curve to exist. Our work generalizes results by \'E. Janvresse, T. de la Rue and Y. Velenik and answers several questions from their work.