# The Density of Numbers Represented by Diagonal Forms of Large Degree

## TL;DR

This paper investigates how the density of integers represented by large-degree diagonal forms decreases rapidly as the degree increases, providing average-case decay estimates.

## Contribution

It establishes that, on average over the degree, the density of numbers represented by fixed diagonal forms diminishes quickly with increasing degree.

## Key findings

- Density decays rapidly with degree k
- Average over k shows significant decline
- Results apply to fixed coefficients a_1,...,a_s

## Abstract

Let $s \geq 3$ be a fixed positive integer and $a_1,\dots,a_s \in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k \] decays rapidly with respect to $k$.