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Additive Diophantine inequalities with mixed powers II

  • Jörg Brüdern (a1)

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A classical problem in the additive theory of numbers is the determination of the minimal s such that for all sufficiently large n the equation

is solvable in natural numbers xk. Improving on earlier results the author [2] has been able to prove that one may take s = 18. In a survey article W. Schwarz asked for an analogue for diophantine inequalities [6]. As a first contribution to this subject we prove

Theorem. Let λ2, …, λ23 be nonzero real numbers, λ23 irrational. Then the values taken by

at integer points ( x1, …, x22) are dense on the real line.

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References

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1.Baker, R. C. and Harman, G.. Diophantine inequalities with mixed powers. J. Number Theory, 18 (1984), 6985.
2.Brüdern, J.. Sums of squares and higher power II. J. Lond. Math. Soc. (2), 35 (1987), 244250.
3.Brüdern, J.. Additive diophantine inequalities with mixed powers I. Mathematika, 34 (1987), 124130.
4.Davenport, H.. Indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81101.
5.Davenport, H. and Heilbronn, H.. On indefinite quadratic forms in five variables. J. Lond. Math. Soc., 21 (1946), 185193.
6.Schwarz, W.. Survey on analogues of the Waring problem: diophantine inequalities. In: Additive Number Theory (Seminaire theorie des nombres, Bordeaux, 1977).
7.Thanigasalam, K.. On sums of powers and a related problem. Acta Arith., 36 (1980), 125141.
8.Vaughan, R. C.. On sums of mixed powers. J. Lond. Math. Soc. (2). 3 (1971), 677688.
9.Vaughan, R. C.. Diophantine approximation by prime numbers II. Proc. Lond. Math Soc. (3), 28 (1974), 385401.
10.Vaughan, R. C.. The Hardy-Littlewood method (Cambridge, 1981).
11.Watson, G. L.. On idenfinite quadratic forms in five variables. Proc. Lond. Math. Soc. (3), 3 (1953), 170181.
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Additive Diophantine inequalities with mixed powers II

  • Jörg Brüdern (a1)

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