Abstract
The purpose of this paper is to present some further applications of the general decoupling theory from [B-D] and [B-D2] to certain diophantine issues. In particular, we consider mean value estimates relevant to the Bombieri- Iwaniec approach to exponential sums and arising in the work of Robert and Sargos [R-S]. Our main input is a new mean-value theorem.
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J. Bourgain and C. Demeter, The proof of the l 2-decoupling conjecture, Ann. of Math. (2), 182 (2015), 351–389.
J. Bourgain and C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, J. Anal.Math., 133 (2017), 279–311.
J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., 21 (2011), 1239–1295.
E. Bombieri and H. Iwaniec, On the order of ξ (1/2 + it), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 449–472.
E. Bombieri and H. Iwaniec, Some mean value theorems for exponential sums, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 473–486.
S. W. Graham and G. Kolesnik, Van der Corput’s Method of Exponential Sums, Cambridge University Press, Cambridge, 1991.
M. N. Huxley, Area, Lattice Points and Exponential Sums, The Clarendon Press, Oxford University Press, New York, 1996.
M. N. Huxley, Exponential sums and the Riemann zeta function IV, Proc. London Math. Soc. (3) 66 (1993), 1–40.
M. N. Huxley, Exponential sums and the Riemann zeta function V, Proc. London Math. Soc. (3) 90 (2005), 1–41.
M. N. Huxley and G. Kolesnik, Exponential sums and the Riemann zeta function III, Proc. London Math. Soc. (3) 62 (1991), 449–468.
S. Parsell, A note on Weyl’s inequality for eighth powers, Rocky Mountain J. Math. 44 (2014), 259–268.
O. Robert and P. Sargos, Un théorème de moyenne pour les sommes d’exponentielles. Application à l’inégatité de Weyl, Publ. Inst. Math. (Beograd) N.S., 67 (2000), 14–30.
T. Wooley, Mean value estimates for odd cubic Weyl sums, Bull. London Math. Soc., 47 (2015), 946–957.
T. Wooley, Translation invariance, exponential sums and Waring’s problem, arXiv:1404.3508v1[math.NT].
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Bourgain, J. Decoupling inequalities and some mean-value theorems. JAMA 133, 313–334 (2017). https://doi.org/10.1007/s11854-017-0035-2
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DOI: https://doi.org/10.1007/s11854-017-0035-2