Duffin–Schaeffer conjecture

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The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941.[1] It states that if ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}}$ is a real-valued function taking on positive values, then for almost all ${\displaystyle \alpha }$ (with respect to Lebesgue measure), the inequality

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

has infinitely many solutions in co-prime integers ${\displaystyle p,q}$ with ${\displaystyle q>0}$ if and only if

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

where ${\displaystyle \varphi (q)}$ is the Euler totient function.

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.[2][3][4]

Progress

The implication from the existence of the rational approximations to the divergence of the series follows from the Borel–Cantelli lemma.[5] The converse implication is the crux of the conjecture.[2] 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 ${\displaystyle c>0}$ such that for every integer ${\displaystyle n}$ we have either ${\displaystyle f(n)=c/n}$ or ${\displaystyle f(n)=0}$.[2][6] This was strengthened by Jeffrey Vaaler in 1978 to the case ${\displaystyle f(n)=O(n^{-1})}$.[7][8] More recently, this was strengthened to the conjecture being true whenever there exists some ${\displaystyle \varepsilon >0}$ such that the series

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {f(n)}{n}}\right)^{1+\varepsilon }\varphi (n)=\infty }$. This was done by Haynes, Pollington, and Velani.[9]

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 is published in the Annals of Mathematics.[10]

In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture.[11][12]

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. ^ a b c Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society. p. 204. ISBN 978-0-8218-0737-8. Zbl 0814.11001.
3. ^ 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. Zbl 0715.11036.
4. ^ Harman (2002) p. 69
5. ^ Harman (2002) p. 68
6. ^ Harman (1998) p. 27
7. ^ "Duffin-Schaeffer Conjecture" (PDF). Ohio State University Department of Mathematics. 2010-08-09. Retrieved 2019-09-19.
8. ^ Harman (1998) p. 28
9. ^ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), https://arxiv.org/abs/0811.1234
10. ^ 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. Zbl 1148.11033.
11. ^ Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593. Cite journal requires |journal= (help)
12. ^ Sloman, Leila (2019). "New Proof Solves 80-Year-Old Irrational Number Problem". Scientific American.