Journal d'Analyse Mathématique

, Volume 115, Issue 1, pp 121–186 | Cite as

Linear forms and quadratic uniformity for functions on ℤN

Article

Abstract

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = Fpn, provided p is not too small. In ℤN, substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm \({\left\| \cdot \right\|_{{U^3}}}\) requires a more sophisticated “local” formulation. When N is prime, ℤN is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.

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Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of Pure mathematics and Mathematical StatisticsCambridgeUK
  2. 2.Department of Mathematics RutgersThe State University of New JerseyPiscatawayUSA

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