# Normal numbers with given limits of multiple ergodic averages

## TL;DR

This paper investigates the properties of normal sequences with specific pattern frequencies, focusing on the topological entropy of sets with prescribed limits of multiple ergodic averages in binary sequences.

## Contribution

It characterizes the topological entropy of sets of normal sequences with fixed pattern frequencies related to multiple ergodic averages.

## Key findings

- Determined the topological entropy for sets with given pattern frequencies.
- Analyzed the structure of normal sequences under multiple ergodic average constraints.

## Abstract

We are interested in the set of normal sequences in the space $\{0,1\}^\mathbb{N}$ with a given frequency of the pattern $11$ in the positions $k, 2k$. The topological entropy of such sets is determined.