Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-17T10:02:33.657Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE SQUARE-FREE REPRESENTATION FUNCTION OF A NORM FORM AND NILSEQUENCES

Part of: Diophantine equations Sequences and sets Multiplicative number theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 November 2015

Lilian Matthiesen
Lilian Matthiesen*
Affiliation:
Institut de Mathématiques de Jussieu – Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France (lilian.matthiesen@imj-prg.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the restriction to square-free numbers of the representation function attached to a norm form does not correlate with nilsequences. By combining this result with previous work of Browning and the author, we obtain an application that is used in recent work of Harpaz and Wittenberg on the fibration method for rational points.

Keywords

norm form equationsweak approximationhigher degree uniformitynilsequences

MSC classification

Primary: 14G05: Rational points 11D57: Multiplicative and norm form equations 11B30: Arithmetic combinatorics; higher degree uniformity 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 1 , February 2018 , pp. 107 - 135
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Browning, T. D. and Matthiesen, L., Norm forms for arbitrary number fields as products of linear polynomials, arXiv:1307.7641 (submitted).Google Scholar
Green, B. J. and Tao, T. C., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481–547.CrossRefGoogle Scholar
Green, B. J. and Tao, T. C., Linear equations in primes, Ann. of Math. (2) 171 (2010), 1753–1850.CrossRefGoogle Scholar
Green, B. J. and Tao, T. C., The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465–540.CrossRefGoogle Scholar
Green, B. J. and Tao, T. C., The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.CrossRefGoogle Scholar
Green, B. J. and Tao, T. C., On the quantitative behaviour of polynomial orbits on nilmanifolds – Erratum, arXiv:1311.6170.Google Scholar
Green, B. J., Tao, T. C. and Ziegler, T., An inverse theorem for the Gowers U s+1[N]-norm, Ann. of Math. (2) 176 (2012), 1231–1372.CrossRefGoogle Scholar
Harpaz, Y. and Wittenberg, O., On the fibration method for zero-cycles and rational points, Ann. of Math. (2), to appear.Google Scholar
Leibman, A., Polynomial mappings of groups (corrected version), Israel J. Math. 129 (2002), 29–60.CrossRefGoogle Scholar
Matthiesen, L., Linear correlations amongst numbers represented by positive definite binary quadratic forms, Acta Arith. 154 (2012), 235–306.CrossRefGoogle Scholar
Matthiesen, L., Generalized Fourier coefficients of multiplicative functions, arXiv:1405.1018.Google Scholar
Matthiesen, L., A consequence of the factorisation theorem for polynomial nilsequences – Corrigendum to Acta Arith. 154 (2012), 235–306, arXiv:1509.06030 (submitted).CrossRefGoogle Scholar