Vol. 14, No. 6, 2020

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Gowers norms control diophantine inequalities

Aled Walker

Vol. 14 (2020), No. 6, 1457–1536

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 fUs+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.

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Gowers norms, diophantine inequalities, Möbius orthogonality, generalised von Neumann theorem
Mathematical Subject Classification 2010
Primary: 11D75
Secondary: 11B30, 11J25
Received: 6 March 2019
Revised: 30 October 2019
Accepted: 6 February 2020
Published: 30 July 2020
Aled Walker
Centre de Recherches Mathématiques
Montréal QC