Diophantine approximation: Difference between revisions

Revision as of 01:52, 10 January 2022

Revision as of 21:18, 26 January 2022

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{{harvtxt|Duffin|Schaeffer|1941}} proved a generalization of Khinchin's result, and posed what is now known as the [[Duffin–Schaeffer conjecture]] on the analogue of Khinchin's dichotomy for general, not necessarily decreasing, sequences <math>\psi</math> . {{harvtxt|Beresnevich|Velani|2006}} proved that a [[Hausdorff measure]] analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker.

In July 2019, Dimitris Koukoulopoulos and [[James Maynard (mathematician)|James Maynard]] announced a proof of the conjecture.<ref>{{cite paperarXiv |firstfirst1=D. |lastlast1=Koukoulopoulos |first2=J. |last2=Maynard |title=On the Duffin–Schaeffer conjecture |year=2019 |arxiv=1907.04593 }}</ref><ref>{{cite journal |last=Sloman |first=Leila |year=2019 |title=New Proof Solves 80-Year-Old Irrational Number Problem |journal=[[Scientific American]] |url=https://www.scientificamerican.com/article/new-proof-solves-80-year-old-irrational-number-problem/ }}</ref>

=== Hausdorff dimension of exceptional sets ===

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==References==

{{refbegin|30em}}

*{{cite journal |zbl=1148.11033 |last1=Beresnevich |first1=Victor |last2=Velani |first2=Sanju |title=A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures |journal=[[Annals of Mathematics]] |volume=164 |issue=3 |year=2006 |pages=971–992 |doi=10.4007/annals.2006.164.971|arxiv=math/0412141 |s2cid=14475449 }}

*{{cite book

| last1 = Bernik | first1 = V.

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*{{cite journal |zbl=0025.11002 |last1=Duffin |first1=R. J. |last2=Schaeffer |first2=A. C. |title=Khintchine's problem in metric diophantine approximation |journal=[[Duke Mathematical Journal]] |volume=8 |issue=2 |pages=243–255 |year=1941 |issn=0012-7094 |doi=10.1215/s0012-7094-41-00818-9}}

*{{cite journal | last1=Dyson | first1=Freeman J. | author1-link=Freeman Dyson | title=The approximation to algebraic numbers by rationals | doi= 10.1007/BF02404697 | mr=0023854 | zbl=0030.02101 | year=1947 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=79 | pages=225–240 | doi-access=free }}

*{{cite book |firstfirst1=G. H. |lastlast1=Hardy |author1-link=G. H. Hardy |first2=E. M. |last2=Wright |author2-link=E. M. Wright |title=An Introduction to the Theory of Numbers |edition=5th |year=1979 |publisher=Oxford University Press |isbn=978-0-19-853170-8 |mr=568909 |title-link=An Introduction to the Theory of Numbers }}

*{{cite journal |first=A. |last=Hurwitz |author-link=Adolf Hurwitz |title=Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche |trans-title=On the approximate representation of irrational numbers by rational fractions |language=de |journal=Mathematische Annalen |volume=39 |year=1891 |issue=2 |pages=279–284 |mr=1510702 |doi=10.1007/BF01206656 |s2cid=119535189 }}

* {{cite book |first=A. Ya. |last=Khinchin |author-link=Aleksandr Khinchin |title=Continued Fractions |publisher=Dover |year=1997 |orig-year=1964 |isbn=0-486-69630-8 }}

*{{cite journal |last1=Kleinbock |first1=D. Y. |last2=Margulis |first2=G. A. |author2-link= Grigory Margulis |title=Flows on homogeneous spaces and Diophantine approximation on manifolds |journal=Ann. Math. |volume=148 |issue=1 |year=1998 |pages=339–360 |mr=1652916 |zbl=0922.11061 |doi=10.2307/120997 |jstor=120997|arxiv=math/9810036 |bibcode=1998math.....10036K |s2cid=8471125 }}

* {{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Introduction to Diophantine Approximations |edition=New expanded |publisher=[[Springer-Verlag]] |year=1995 |isbn=0-387-94456-7 |zbl=0826.11030 }}

* {{cite book |first=G. A. |last=Margulis |author-link=Grigory Margulis |chapter=Diophantine approximation, lattices and flows on homogeneous spaces |title=A panorama of number theory or the view from Baker's garden |editor1-last=Wüstholz |editor1-first=Gisbert |editor1-link=Gisbert Wüstholz |pages=280–310 |publisher=[[Cambridge University Press]] |___location=Cambridge |year=2002 |mr=1975458 |isbn=0-521-80799-9 }}

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* {{cite book |zbl=0421.10019 |last=Schmidt |first=Wolfgang M. |author-link=Wolfgang M. Schmidt |edition=1996 |title=Diophantine approximation |series=Lecture Notes in Mathematics |volume=785 |___location=Berlin-Heidelberg-New York |publisher=[[Springer-Verlag]] |year=1980 |isbn=3-540-09762-7 }}

* {{cite book |last=Schmidt |first=Wolfgang M. |author-link=Wolfgang M. Schmidt |title=Diophantine approximations and Diophantine equations |series=Lecture Notes in Mathematics |volume=1467 |publisher=[[Springer-Verlag]] |year=1996 |edition=2nd |isbn=3-540-54058-X |zbl=0754.11020 }}

*{{cite journal | last1=Siegel | first1=Carl Ludwig | author1-link=Carl Ludwig Siegel | title=Approximation algebraischer Zahlen | doi=10.1007/BF01211608 | year=1921 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=10 | issue=3 | pages=173–213 | s2cid=119577458 | url=https://zenodo.org/record/1538156 }}

* {{cite book |last=Sprindzhuk |first=Vladimir G. |title=Metric theory of Diophantine approximations |others=Transl. from the Russian and ed. by Richard A. Silverman. With a foreword by Donald J. Newman |series=Scripta Series in Mathematics |publisher=John Wiley & Sons |year=1979 |isbn=0-470-26706-2 |mr=0548467 |zbl=0482.10047}}

*{{cite journal | last1=Thue | first1=A. | author1-link=Axel Thue | title=Über Annäherungswerte algebraischer Zahlen | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0135 | year=1909 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1909 | issue=135 | pages=284–305 | doi=10.1515/crll.1909.135.284 | s2cid=125903243 }}

{{refend}}