Duffin–Schaeffer theorem

Mathematical theorem

The Duffin–Schaeffer theorem is a theorem in mathematics, specifically, the Diophantine approximation proposed as a conjecture by R. J. Duffin and A. C. Schaeffer in 1941[1] and proven in 2019 by Dimitris Koukoulopoulos and James Maynard.[2] 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.[3][4][5]

Introduction

That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma.[6] The converse implication is the crux of the conjecture.[3] 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} .[3][7] This was strengthened by Jeffrey Vaaler in 1978 to the case f ( n ) = O ( n − 1 ) {\displaystyle f(n)=O(n^{-1})} .[8][9] 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.[10]

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.[11]

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. ^ a b c 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

  • Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. Vol. 18. Oxford: Clarendon Press. ISBN 978-0-19-850083-4. Zbl 1081.11057.
  • Harman, Glyn (2002). "One hundred years of normal numbers". In Bennett, M. A.; Berndt, B.C.; Boston, N.; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. (eds.). Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 57–74. ISBN 978-1-56881-162-8. Zbl 1062.11052.

External links

  • Quanta magazine article about Duffin-Schaeffer conjecture.
  • Numberphile interview with James Maynard about the proof.