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ON A DIAGONAL QUADRIC IN DENSE VARIABLES

Published online by Cambridge University Press:  22 August 2014

EUGEN KEIL
EUGEN KEIL*
Affiliation:
Howard House, Department of Mathematics, University Walk, Clifton, Bristol BS8 1TW, United Kingdom e-mails: maxek@bristol.ac.ukMathematical Institute, 24-29 St Giles', Oxford OX1 3LB, United Kingdom e-mails: Eugen.Keil@maths.ox.ac.uk
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Abstract

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We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets $\mathcal{A}$ ⊂ ℤ and show quantitative bounds on the size of $\mathcal{A}$ if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s ≥ 7 variables.

Keywords

Primary 11B30Secondary 11P5511D0911L07

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014, pp. 601 - 628
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Baker, R. C., Diophantine inequalities (Clarendon Press, Oxford University Press, New York, NY, 1986).Google Scholar
2.Bergelson, V. and Leibman, A., Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc. 9 (3) (1996), 725–753.CrossRefGoogle Scholar
3.Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (2) (1993), 107–156.Google Scholar
4.Bourgain, J., Roth's theorem on progressions revisited, J. Anal. Math. 104 (2008), 155–192.Google Scholar
5.Erdős, P. and Turan, P.On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.CrossRefGoogle Scholar
6.Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.CrossRefGoogle Scholar
7.Furstenberg, H., Katznelson, Y. and Ornstein, D., The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (NS) 7 (3) (1982), 527–552.Google Scholar
8.Gowers, T., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (3) (2001), 465–588.CrossRefGoogle Scholar
9.Green, B., On arithmetic structures in dense sets of integers, Duke Math. J. 114 (2) (2002), 215–238.Google Scholar
10.Green, B., Roth's theorem in the primes, Ann. Math. (2) 161 (3) (2005), 1609–1636.CrossRefGoogle Scholar
11.Green, B. and Tao, T., Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux 18 (1) (2006), 147–182.Google Scholar
12.Green, B. and Tao, T., Linear equations in primes, Ann. Math. 171 (3) (2010), 1753–1850.CrossRefGoogle Scholar
13.Greenberg, M. J., Lectures on forms in many variables (W. A. Benjamin, New York, NY 1969).Google Scholar
14.Heath-Brown, D. R., Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (3) (1987), 385–394.CrossRefGoogle Scholar
15.Huxley, M. N., Area, lattice points, and exponential sums (Clarendon Press, Oxford University Press, New York, NY, 1996).CrossRefGoogle Scholar
16.Meyer, A., Über die Auflösung der Gleichung ax 2 + by 2 + cz 2 + du 2 + ev 2 = 0 in ganzen Zahlen, Vierteljahrsschr. Naturforsch. Ges. Zürich 29 (1884), 209–222.Google Scholar
17.Rogovskaya, N. N., An asymptotic formula for the number of solutions of a system of equations, in Diophantine approximations, Part II (Russian) (Moskov. Gos. Univ., Moscow, Russia, 1986), 78–84.Google Scholar
18.Roth, K. F., On certain sets of integers, J. London Math. Soc. 28, (1953), 104–109.Google Scholar
19.Sanders, T., On Roth's theorem on progressions, Ann. Math. 174 (1) (2011), 619–636.Google Scholar
20.Sárközy, A., On difference sets of sequences of integers, I. Acta Math. Acad. Sci. Hungary 31 (1–2), (1978), 125–149.CrossRefGoogle Scholar
21.Smith, M., On solution-free sets for simultaneous quadratic and linear equations, J. London Math. Soc. (2) 79 (2) (2009), 273–293.Google Scholar
22.Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.Google Scholar
23.Vaughan, R. C., The Hardy–Littlewood method, 2nd ed. (Cambridge University Press, Cambridge, UK, 1997).Google Scholar
24.Zaharescu, A., Small values of n 2α (mod 1), Invent. Math. 121 (2) (1995), 379–388.Google Scholar