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The '''Duffin–Schaeffer conjecture''' is an important conjecture in [[metric number theory]] proposed by [[Richard Duffin|R. J. Duffin]] and [[Albert Charles Schaeffer|A. C. Schaeffer]] in 1941.<ref>R. J. Duffin and A. C. Schaeffer, ''Khintchine's problem in metric [[Diophantine approximation]]'', [[Duke Mathematical Journal]], '''8''' (1941), 243–255</ref> It states that if <math>f : \mathbb{N} \rightarrow \mathbb{R}^+</math> is a real-valued function taking on positive values, then for [[almost all]] <math> \alpha </math> (with respect to [[Lebesgue measure]]), the inequality | |||
: <math> \left | \alpha - \frac{p}{q} \right | < \frac{f(q)}{q} </math> | |||
has infinitely many solutions in [[co-prime]] integers <math>p,q</math> with <math>q > 0</math> if and only if the sum | |||
: <math> \sum_{q=1}^\infty f(q) \frac{\varphi(q)}{q} = \infty. </math> | |||
Here <math> \varphi(q)</math> is the [[Euler totient function]]. | |||
The full conjecture remains unsolved. However, a higher dimensional analogue of this conjecture has been resolved.<ref name=Mon204/><ref>{{cite journal | first1=A.D. | last1=Pollington | first2=R.C. | last2=Vaughan | author2-link=Bob Vaughan | title=The ''k'' dimensional Duffin–Schaeffer conjecture | journal=Mathematika | volume=37 | number=2 | year=1990 | pages=190–200 | zbl=0715.11036 | issn=0025-5793 }}</ref><ref name=Har0269>Harman (2002) p.69</ref> | |||
==Progress== | |||
The implication from the existence of the rational approximations to the divergence of the series follows from the [[Borel–Cantelli lemma]].<ref name=Har0268>Harman (2002) p.68</ref> The converse implication is the crux of the conjecture.<ref name=Mon204/> | |||
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 <math>c > 0 </math> such that for every integer <math> n </math> we have either <math>f(n) = c/n</math> or <math>f(n) = 0</math>.<ref name=Mon204>{{ cite book | last=Montgomery | first=Hugh L. | author-link=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 | page=204 }}</ref><ref name=Har27>Harman (1998) p.27</ref> This was strengthened by Jeffrey Vaaler in 1978 to the case <math>f(n) = O(n^{-1})</math>.<ref>http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf</ref><ref name=Har28>Harman (1998) p.28</ref> More recently, this was strengthened to the conjecture being true whenever there exists some <math>\epsilon > 0 </math> such that the series <math>\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \varphi(n) = \infty </math>. This was done by Haynes, Pollington, and Velani.<ref>A. Haynes, A. Pollington, and S. Velani, ''The Duffin-Schaeffer Conjecture with extra divergence'', arXiv, (2009), http://arxiv.org/abs/0811.1234</ref> | |||
In 2006, Beresnevich and Velani proved that a [[Hausdorff dimension]] 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]]''.<ref>{{cite journal | 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]] (2) | volume=164 | number=3 | year=2006 | pages=971–992 | url=http://arxiv.org/abs/math/0412141 | zbl=1148.11033 | issn=0003-486X }}</ref> | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
* {{cite book | last=Harman | first=Glyn | author-link= | title=Metric number theory | series=London Mathematical Society Monographs. New Series | volume=18 | location=Oxford | publisher=[[Clarendon Press]] | year=1998 | isbn=0-19-850083-1 | zbl=1081.11057 }} | |||
* {{cite book | last=Harman | first=Glyn | authorlink=Glyn Harman | chapter=One hundred years of normal numbers | editor1-last=Bennett | editor1-first=M. A. | editor2-last=Berndt | editor2-first=B.C. | editor2-link=Bruce C. Berndt | editor3-last=Boston | editor3-first=N. | editor3-link=Nigel Boston | editor4-last=Diamond | editor4-first=H.G. | editor5-last=Hildebrand | editor5-first=A.J. | editor6-last=Philipp | editor6-first=W. | title=Surveys in number theory: Papers from the millennial conference on number theory | location=Natick, MA | publisher=A K Peters | pages=57–74 | year=2002 | isbn=1-56881-162-4 | zbl=1062.11052 }} | |||
{{DEFAULTSORT:Duffin-Schaeffer conjecture}} | |||
[[Category:Conjectures]] | |||
[[Category:Diophantine approximation]] | |||
Revision as of 02:13, 22 October 2013
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 is a real-valued function taking on positive values, then for almost all (with respect to Lebesgue measure), the inequality
has infinitely many solutions in co-prime integers with if and only if the sum
Here is the Euler totient function.
The full conjecture remains unsolved. However, a higher dimensional analogue of this conjecture has been resolved.[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 such that for every integer we have either or .[2][6] This was strengthened by Jeffrey Vaaler in 1978 to the case .[7][8] More recently, this was strengthened to the conjecture being true whenever there exists some such that the series . This was done by Haynes, Pollington, and Velani.[9]
In 2006, Beresnevich and Velani proved that a Hausdorff dimension 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]
Notes
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References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Mathematical Journal, 8 (1941), 243–255
- ↑ 2.0 2.1 2.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Harman (2002) p.69
- ↑ Harman (2002) p.68
- ↑ Harman (1998) p.27
- ↑ http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf
- ↑ Harman (1998) p.28
- ↑ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), http://arxiv.org/abs/0811.1234
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang