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    Cochrane, Todd and Shi, Sanying 2010. The congruence <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mspace width="0.25em"/><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">mod</mml:mi><mml:mspace width="0.25em"/><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and mean values of character sums. Journal of Number Theory, Vol. 130, Issue. 3, p. 767.


    Chan, Tsz Ho 2008. Approximating reals by sums of two rationals. Journal of Number Theory, Vol. 128, Issue. 5, p. 1182.


    Garaev, M.Z. and Garcia, V.C. 2008. The equation <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>λ</mml:mi></mml:math> in fields of prime order and applications. Journal of Number Theory, Vol. 128, Issue. 9, p. 2520.


    Harman, Glyn 1988. Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 103, Issue. 02, p. 197.


    Harman, Glyn 1985. Diophantine approximation with almost-primes and sums of two squares. Mathematika, Vol. 32, Issue. 02, p. 301.


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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 95, Issue 3
  • May 1984, pp. 381-388

Diophantine approximation with square-free integers

  • Glyn Harman (a1)
  • DOI: http://dx.doi.org/10.1017/S0305004100061685
  • Published online: 24 October 2008
Abstract

In this paper we shall prove the following two results.

Theorem 1. Let ∊ > 0 and β a real number be given. Them, for almost all real a (in the sense of Lebesque measure), there are infinitely many pairs of square-free integers m, n such that

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[1]H. Davenport . Multiplicative Number Theory, 2nd ed. Revised by H. L. Montgomery . (Springer-Verlag, 1980).

[11]H. L. Montgomery . Topics in Multiplicative Number Theory. (Springer-Verlag, 1971).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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