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  • Sam Chow (a1)


Let ${\it\mu}_{1},\ldots ,{\it\mu}_{s}$ be real numbers, with ${\it\mu}_{1}$ irrational. We investigate sums of shifted $k$ th powers $\mathfrak{F}(x_{1},\ldots ,x_{s})=(x_{1}-{\it\mu}_{1})^{k}+\cdots +(x_{s}-{\it\mu}_{s})^{k}$ . For $k\geqslant 4$ , we bound the number of variables needed to ensure that if ${\it\eta}$ is real and ${\it\tau}>0$ is sufficiently large then there exist integers $x_{1}>{\it\mu}_{1},\ldots ,x_{s}>{\it\mu}_{s}$ such that $|\mathfrak{F}(\mathbf{x})-{\it\tau}|<{\it\eta}$ . This is a real analogue to Waring’s problem. When $s\geqslant 2k^{2}-2k+3$ , we provide an asymptotic formula. We prove similar results for sums of general univariate degree- $k$ polynomials.



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1.Baker, R. C., Diophantine Inequalities (London Mathematical Society Monographs (N.S.) 1), Clarendon Press (Oxford, 1986).
2.Brüdern, J., Kawada, K. and Wooley, T. D., Additive representation in thin sequences, VIII: Diophantine inequalities in review. In Number Theory, 20–79 (Series on Number Theory and its Applications 6), World Scientific (Hackensack, NJ, 2010).
3.Chow, S., Cubic diophantine inequalities for split forms. Monatsh. Math. 175 2014, 213225.
4.Chow, S., Sums of cubes with shifts. J. Lond. Math. Soc. (2) 2015, doi:10.1112/jlms/jdu077.
5.Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edn., Cambridge University Press (Cambridge, 2005).
6.Davenport, H. and Heilbronn, H., On indefinite quadratic forms in five variables. J. Lond. Math. Soc. 21 1946, 185193.
7.Freeman, D. E., One cubic diophantine inequality. J. Lond. Math. Soc. (2) 61(1) 2000, 2535.
8.Freeman, D. E., Asymptotic lower bounds and formulas for Diophantine inequalities. In Number Theory for the Millennium II, A. K. Peters (Natick, MA, 2002), 5774.
9.Freeman, D. E., Additive inhomogeneous Diophantine inequalities. Acta Arith. 107(3) 2003, 209244.
10.Götze, F., Lattice point problems and values of quadratic forms. Invent. Math. 157(1) 2004, 195226.
11.Hardy, G. H. and Littlewood, J. E., Some problems of “Partitio numerorum” (VI): further researches in Waring’s problem. Math. Z. 23(1) 1925, 137.
12.Harvey, M. P., Cubic Diophantine inequalities involving a norm form. Int. J. Number Theory 7(8) 2011, 22192235.
13.Margulis, G. A., Discrete subgroups and ergodic theory. In Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press (Boston, MA, 1989), 377398.
14.Margulis, G. A. and Mohammadi, A., Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. Duke Math. J. 158(1) 2011, 121160.
15.Marklof, J., Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math. (2) 158(2) 2003, 419471.
16.Nathanson, M. B., Elementary Methods in Number Theory (Graduate Texts in Mathematics 195), Springer (New York, 2000).
17.Parsell, S. T. and Wooley, T. D., Exceptional sets for Diophantine inequalities. Int. Math. Res. Not. IMRN 2014(14) 2014, 39193974.
18.Schmidt, W. M., Diophantine inequalities for forms of odd degree. Adv. Math. 38(2) 1980, 128151.
19.Vaughan, R. C., On Waring’s problem for sixth powers. J. Lond. Math. Soc. (2) 33(2) 1986, 227236.
20.Vaughan, R. C., On Waring’s problem for smaller exponents. Proc. Lond. Math. Soc. (3) 52(3) 1986, 445463.
21.Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn., Cambridge University Press (Cambridge, 1997).
22.Vaughan, R. C. and Wooley, T. D., Waring’s problem: a survey. In Number Theory for the Millennium III, A. K. Peters (Natick, MA, 2002), 301340.
23.Waring, E., Meditationes Arithmeticæ, 2nd edn., Archdeacon (Cambridge, 1770).
24.Wooley, T. D., On Diophantine inequalities: Freeman’s asymptotic formulae. In Proceedings of the Session in Analytic Number Theory and Diophantine Equations (Bonner Mathematische Schriften 360), University of Bonn (Bonn, 2003).
25.Wooley, T. D., Multigrade efficient congruencing and Vinogradov’s mean value theorem (submitted), Preprint, 2013, arXiv:1310.8447.
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