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Duffin–Schaeffer theorem

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(Redirected from Duffin–Schaeffer conjecture) Mathematical theorem

The Koukoulopoulos–Maynard theorem, also known as the Duffin-Schaeffer conjecture, is a theorem in mathematics, specifically, the Diophantine approximation proposed as a conjecture by R. J. Duffin and A. C. Schaeffer in 1941 and proven in 2019 by Dimitris Koukoulopoulos and James Maynard. It states that if f : N → R + {\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}} is a real-valued function taking on positive values, then for almost all α {\displaystyle \alpha } (with respect to Lebesgue measure), the inequality

| α − p q | < f ( q ) q {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {f(q)}{q}}}

has infinitely many solutions in coprime integers p , q {\displaystyle p,q} with q > 0 {\displaystyle q>0} if and only if

∑ q = 1 ∞ φ ( q ) f ( q ) q = ∞ , {\displaystyle \sum _{q=1}^{\infty }\varphi (q){\frac {f(q)}{q}}=\infty ,}

where φ ( q ) {\displaystyle \varphi (q)} is Euler's totient function.

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.

Introduction

That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma. The converse implication is the crux of the conjecture. There have been many partial results of the Duffin–Schaeffer conjecture established to date. Paul Erdős established in 1970 that the conjecture holds if there exists a constant c > 0 {\displaystyle c>0} such that for every integer n {\displaystyle n} we have either f ( n ) = c / n {\displaystyle f(n)=c/n} or f ( n ) = 0 {\displaystyle f(n)=0} . This was strengthened by Jeffrey Vaaler in 1978 to the case f ( n ) = O ( n − 1 ) {\displaystyle f(n)=O(n^{-1})} . More recently, this was strengthened to the conjecture being true whenever there exists some ε > 0 {\displaystyle \varepsilon >0} such that the series

∑ n = 1 ∞ ( f ( n ) n ) 1 + ε φ ( n ) = ∞ . {\displaystyle \sum _{n=1}^{\infty }\left({\frac {f(n)}{n}}\right)^{1+\varepsilon }\varphi (n)=\infty .}

This was done by Haynes, Pollington, and Velani.

In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result was published in the Annals of Mathematics.

See also

Notes

  1. Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine's problem in metric diophantine approximation". Duke Math. J. 8 (2): 243–255. doi:10.1215/S0012-7094-41-00818-9. JFM 67.0145.03. Zbl 0025.11002.
  2. Koukoulopoulos, Dimitris; Maynard, James (2020). "On the Duffin-Schaeffer conjecture". Annals of Mathematics. 192 (1): 251. arXiv:1907.04593. doi:10.4007/annals.2020.192.1.5. JSTOR 10.4007/annals.2020.192.1.5. S2CID 195874052.
  3. ^ Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. p. 204. ISBN 978-0-8218-0737-8. Zbl 0814.11001.
  4. Pollington, A.D.; Vaughan, R.C. (1990). "The k dimensional Duffin–Schaeffer conjecture". Mathematika. 37 (2): 190–200. doi:10.1112/s0025579300012900. ISSN 0025-5793. S2CID 122789762. Zbl 0715.11036.
  5. Harman (2002) p. 69
  6. Harman (2002) p. 68
  7. Harman (1998) p. 27
  8. "Duffin-Schaeffer Conjecture" (PDF). Ohio State University Department of Mathematics. 2010-08-09. Retrieved 2019-09-19.
  9. Harman (1998) p. 28
  10. A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), https://arxiv.org/abs/0811.1234
  11. Beresnevich, Victor; Velani, Sanju (2006). "A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures". Annals of Mathematics. Second Series. 164 (3): 971–992. arXiv:math/0412141. doi:10.4007/annals.2006.164.971. ISSN 0003-486X. S2CID 14475449. Zbl 1148.11033.

References

External links

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