Abstract
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ℓ 2-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ℓ p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.
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References
M. Avdispahić and L. Smajlović, On maximal operators on k-spheres in Zn, Proceedings of the American Mathematical Society 134 (2006), 2125–2130 (electronic).
J. Bourgain, Estimations de certaines fonctions maximales, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 301 (1985), 499–502.
J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel Journal of Mathematics 61 (1988), 39–72.
J. Bourgain, On the pointwise ergodic theorem on L p for arithmetic sets, Israel Journal of Mathematics 61 (1988), 73–84.
J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Institut des Hautes Études Scientifiques. Publications Mathématiques 69 (1989), 5–45.
M. Cowling and G. Mauceri, Inequalities for some maximal functions. II, Transactions of the American Mathematical Society 296 (1986), 341–365.
H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005.
G. H. Hardy, Collected Papers of G. H. Hardy. Vol. I–VII, The Clarendon Press Oxford University Press, New York, 1979.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford University Press, Oxford, 1998.
Y. Hu and X. Li, Discrete fourier restriction associated with Schrödinger equations, Revista Matemática Iberoamericana 30 (2014), 1281–1300.
I. A. Ikromov, M. Kempe and D. Müller, Estimates for maximal functions associated with hypersurfaces in R 3 and related problems of harmonic analysis, Acta Mathematica 204 (2010), 151–271.
A. D. Ionescu, An endpoint estimate for the discrete spherical maximal function, Proceedings of the American Mathematical Society 132 (2004), 1411–1417 (electronic).
A. Magyar, L p-bounds for spherical maximal operators on Z n, Revista Matemática Iberoamericana 13 (1997), 307–317.
A. Magyar, Diophantine equations and ergodic theorems, American Journal of Mathematics 124 (2002), 921–953.
A. Magyar, On the distribution of lattice points on spheres and level surfaces of polynomials, Journal of Number Theory 122 (2007), 69–83.
A. Magyar, E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis: spherical averages, Annals of Mathematics 155 (2002), 189–208.
A. Magyar, On distance sets of large sets of integer points, Israel Journal of Mathematics 164 (2008), 251–263.
H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1994.
E. M. Stein, Maximal functions. I. Spherical means, Proceedings of the National Academy of Sciences of the United States of America 73 (1976), 2174–2175.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003.
R. C. Vaughan, The Hardy–Littlewood Method, second edition, Cambridge Tracts in Mathematics, Vol. 125, Cambridge University Press, Cambridge, 1997.
T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Annals of Mathematics 175 (2012), 1575–1627.
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Hughes, K. Maximal functions and ergodic averages related to Waring’s problem. Isr. J. Math. 217, 17–55 (2017). https://doi.org/10.1007/s11856-017-1437-7
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DOI: https://doi.org/10.1007/s11856-017-1437-7