Random Diophantine inequalities of additive type

TL;DR

This paper proves that for most additive Diophantine inequalities of degree k with more than 2k variables, the expected number of solutions matches the asymptotic prediction, marking a first in metric results.

Contribution

It introduces the first metric result for Diophantine inequalities using the Davenport-Heilbronn circle method, establishing the asymptotic density for almost all cases.

Findings

Asymptotic formula holds for almost all inequalities

Results apply to inequalities of degree k with >2k variables

First metric result in the field

Abstract

Using the Davenport-Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree $k$ in more than $2k$ variables the expected asymptotic formula for the density of solutions holds true. This appears to be the first metric result on Diophantine inequalities.