Gowers norms control diophantine inequalities

Walker, Aled

doi:10.2140/ant.2020.14.1457

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Gowers norms control diophantine inequalities

Aled Walker

Vol. 14 (2020), No. 6, 1457–1536

DOI: 10.2140/ant.2020.14.1457

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 $∥ f ∥_{U^{s + 1} [N]}$ 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 Möbius 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.

Keywords

Gowers norms, diophantine inequalities, Möbius orthogonality, generalised von Neumann theorem

Mathematical Subject Classification 2010

Primary: 11D75

Secondary: 11B30, 11J25

Milestones

Received: 6 March 2019

Revised: 30 October 2019

Accepted: 6 February 2020

Published: 30 July 2020

Authors

Aled Walker
Centre de Recherches Mathématiques Montréal QC Canada

Vol. 14, No. 6, 2020