# Gowers norms control diophantine inequalities

## TL;DR

This paper extends the use of Gowers norms from linear equations to inequalities, providing bounds on solutions and conditions for their applicability, with implications for number theory and prime-related problems.

## Contribution

It introduces a framework to control solutions of diophantine inequalities using Gowers norms and establishes necessary and sufficient conditions for this control.

## Key findings

- Gowers norms bound solutions to diophantine inequalities
- A characterization of systems governed by Gowers norms
- Applications to M"{o}bius function cancellation

## Abstract

A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the M\"{o}bius function over certain sequences. The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues.