# Local limit theorem in deterministic systems

## TL;DR

This paper proves that in any ergodic and aperiodic deterministic system, one can construct a function whose partial sums follow a lattice local limit theorem, extending classical probabilistic results to deterministic dynamics.

## Contribution

It establishes the existence of functions in deterministic systems whose partial sums obey the lattice local limit theorem, bridging probabilistic limit theorems with dynamical systems.

## Key findings

- Existence of functions satisfying the lattice local limit theorem in ergodic systems
- Extension of probabilistic limit theorems to deterministic systems
- Provides a new link between ergodic theory and probability

## Abstract

We show that for every ergodic and aperiodic probability preserving system, there exists a $\mathbb{Z}$ valued, square integrable function $f$ such that the partial sums process of the time series $\left\{f\circ T^i\right\}_{i=0}^\infty$ satisfies the lattice local limit theorem.