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

  • Lilian Matthiesen (a1)
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.

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1. Browning, T. D. and Matthiesen, L., Norm forms for arbitrary number fields as products of linear polynomials, arXiv:1307.7641 (submitted).
2. Green, B. J. and Tao, T. C., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481547.
3. Green, B. J. and Tao, T. C., Linear equations in primes, Ann. of Math. (2) 171 (2010), 17531850.
4. Green, B. J. and Tao, T. C., The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465540.
5. Green, B. J. and Tao, T. C., The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541566.
6. Green, B. J. and Tao, T. C., On the quantitative behaviour of polynomial orbits on nilmanifolds – Erratum, arXiv:1311.6170.
7. 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), 12311372.
8. Harpaz, Y. and Wittenberg, O., On the fibration method for zero-cycles and rational points, Ann. of Math. (2), to appear.
9. Leibman, A., Polynomial mappings of groups (corrected version), Israel J. Math. 129 (2002), 2960.
10. Matthiesen, L., Linear correlations amongst numbers represented by positive definite binary quadratic forms, Acta Arith. 154 (2012), 235306.
11. Matthiesen, L., Generalized Fourier coefficients of multiplicative functions, arXiv:1405.1018.
12. Matthiesen, L., A consequence of the factorisation theorem for polynomial nilsequences – Corrigendum to Acta Arith. 154 (2012), 235–306, arXiv:1509.06030 (submitted).
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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