# Level set estimates for the discrete frequency function

## TL;DR

This paper introduces the discrete frequency function as a novel tool for analyzing the discrete Hardy-Littlewood maximal function, focusing on its definition, properties, and potential insights into its behavior.

## Contribution

It defines the discrete frequency function, explores its well-definedness, and studies its size and smoothness properties, offering a new perspective on the maximal function.

## Key findings

- Discrete frequency function is well-defined.
- Size and smoothness properties are characterized.
- Provides a new approach to understanding the maximal function.

## Abstract

We introduce the discrete frequency function as a possible new approach to understanding the discrete Hardy-Littlewood maximal function. Considering that the discrete Hardy-Littlewood maximal function is given at each integer by the supremum of averages over intervals of integer length, we define the discrete frequency function at that integer as the value at which the supremum is attained. After verifying that the function is well-defined, we investigate size and smoothness properties of this function.