# Theory of badly approximable sets

Lead Research Organisation:
Durham University

Department Name: Mathematical Sciences

### Abstract

A given real number can be approximated by rational numbers with an arbitrary precision. For example the decimal representation of a number provides such an approximation. On the other hand different numbers are approximated by rationals with different speed in terms of the denominators of that rational numbers. There is no upper bound for that speed which was firstly mentioned by Liouville in 1840's. On the other hand there is a natural lower bound for that speed which is determined by Dirichlet-Hurwitz theorems. It says that for any irrational number x one can find infinitely many rational numbers p/q such that

|x - p/q| < 1/q^2.

Moreover there exist irrational numbers which make this inequality sharp. In other words for that numbers the distance |x-p/q| can not be made smaller than c/q^2 for some positive constant c and all rational numbers p/q. Irrational real numbers with such property are called badly approximable.

The set of badly approximable numbers can be described quite well with help of the theory of continued fractions. It allows us to derive a lot of information about that set: it's "size" in terms of Lebesgue's measure and Hausdorff dimension and some other information about its structure.

In two-dimensional case and more generally in high dimensional case we can also approximate any point in R^n by points with rational coefficients. However different coordinates can be approximated by rationals with their own speed. Because of this phenomena, in higher dimensions we have an uncountable family of different sets of badly approximable points depending on the relative speed of approximation of different coordinates of the point.

The structure of the sets of badly approximable points in high dimensions is much less known than their one-dimensional analogue. It incorporates a lot of open problems. Some of them like famous Littlewood conjecture attract a lot of attention of the modern mathematical society. In this project I am going to shed the light on many of that problems.

One of the main objectives of this project is to construct the mechanism which, like continued fractions in one dimension, will help us to deal with badly approximable points easier. This work has already been started by me together with Velani and Pollington. We described the sets which allowed us to solve a number of related problems including the famous Schmidt conjecture.

Next, the mechanism of generalized Cantor sets will be used to solve some particular open problems about badly approximable points in n-dimensional real space. In particular the structure of BAD points on manifolds in arbitrary dimensions will be investigated.

Finally some new approaches will be applied to attack the famous Littlewood conjecture. It was stated by Littlewood in 1931 and nowadays it is one of the most attractive problems in modern Mathematics.

|x - p/q| < 1/q^2.

Moreover there exist irrational numbers which make this inequality sharp. In other words for that numbers the distance |x-p/q| can not be made smaller than c/q^2 for some positive constant c and all rational numbers p/q. Irrational real numbers with such property are called badly approximable.

The set of badly approximable numbers can be described quite well with help of the theory of continued fractions. It allows us to derive a lot of information about that set: it's "size" in terms of Lebesgue's measure and Hausdorff dimension and some other information about its structure.

In two-dimensional case and more generally in high dimensional case we can also approximate any point in R^n by points with rational coefficients. However different coordinates can be approximated by rationals with their own speed. Because of this phenomena, in higher dimensions we have an uncountable family of different sets of badly approximable points depending on the relative speed of approximation of different coordinates of the point.

The structure of the sets of badly approximable points in high dimensions is much less known than their one-dimensional analogue. It incorporates a lot of open problems. Some of them like famous Littlewood conjecture attract a lot of attention of the modern mathematical society. In this project I am going to shed the light on many of that problems.

One of the main objectives of this project is to construct the mechanism which, like continued fractions in one dimension, will help us to deal with badly approximable points easier. This work has already been started by me together with Velani and Pollington. We described the sets which allowed us to solve a number of related problems including the famous Schmidt conjecture.

Next, the mechanism of generalized Cantor sets will be used to solve some particular open problems about badly approximable points in n-dimensional real space. In particular the structure of BAD points on manifolds in arbitrary dimensions will be investigated.

Finally some new approaches will be applied to attack the famous Littlewood conjecture. It was stated by Littlewood in 1931 and nowadays it is one of the most attractive problems in modern Mathematics.

### Planned Impact

The research proposed in this application primarily aims at developing new techniques and solving outstanding problems in the theory of Diophantine approximation. For decades this area of

research has been a UK strength and has led to numerous breakthroughs, some of them being acknowledged by Fields medals (Roth, Baker). Nowadays the UK position in Diophantine approximation as well as in other research areas particularly close to the proposed research such that ergodic theory and dynamical systems are strong and represented across the country, e.g. at Bristol, Manchester, Warwick, York and other places. Doubtless the UK will continue to hold one of the world-leading positions in the area. The proposed research will help maintaining the UK leadership and contribute to the development of its research capability in this research area. This will highly likely remain important over the next 10-50 years as the various high-profile research activities in the area have only recently started unfolding.

The proposed topics of the research are also very well recognized by the international mathematical society. So the mathematical society on both national and international levels will certainly benefit from this project.

research has been a UK strength and has led to numerous breakthroughs, some of them being acknowledged by Fields medals (Roth, Baker). Nowadays the UK position in Diophantine approximation as well as in other research areas particularly close to the proposed research such that ergodic theory and dynamical systems are strong and represented across the country, e.g. at Bristol, Manchester, Warwick, York and other places. Doubtless the UK will continue to hold one of the world-leading positions in the area. The proposed research will help maintaining the UK leadership and contribute to the development of its research capability in this research area. This will highly likely remain important over the next 10-50 years as the various high-profile research activities in the area have only recently started unfolding.

The proposed topics of the research are also very well recognized by the international mathematical society. So the mathematical society on both national and international levels will certainly benefit from this project.

## People |
## ORCID iD |

Dzmitry Badziahin (Principal Investigator) |

### Publications

Badziahin D
(2015)

*An unusual continued fraction*in Proceedings of the American Mathematical Society
Badziahin D

*On Generalised Thue-Morse functions and their values*in Mathematical Proceedings of the Cambridge Philosophical Society
Badziahin D
(2015)

*Thue-Morse Constant is Not Badly Approximable*in International Mathematics Research Notices
Badziahin D

*An inhomogeneous Jarnik type theorem for planar curves*in Mathematical Proceedings of the Cambridge Philosophical society
Badziahin D
(2015)

*On the complexity of a putative counterexample to the -adic Littlewood conjecture*in Compositio Mathematica
Badziahin D

*Cantor-winning sets and their applications*in Mathematische Annalen
Badziahin D
(2017)

*Cantor-winning sets and their applications*in Advances in Mathematics
Badziahin D
(2015)

*Computation of the Infimum in the Littlewood Conjecture*in Experimental MathematicsDescription | I managed to develop a theory of badly approximable points. It enabled me to solve several open problems which were in the focus of specialists in Diophantine Approximations. As a result one paper was written by me together with Stephen Harrap, my postdoc and submitted to the journal (Math. Proc. Camb. Phil. Soc.). Another paper on this topic is under way and will be accomplished soon. Concerning another project objective - Littlewood conjecture - I together with my colleagues Bugeaud, Einsiedler and Kleinbock managed to significantly advance in solving the p-adic analogue of it. Recently I submitted three papers on this topic. Two of them are already accepted. One in Compositio Math and another one in Experimental Math. The other one is still pending. I completed several small related projects with my collaborators: 1) Together with Zorin I discovered some approximational properties of Thue-Morse number and some related numbers. One our paper on this topic is published in IMRN and another one is submitted. 2) Together with Hussain and Harrap we improved the Khintchine-Groshev-Jarnik type result for well approximable points on planar curves. As a result we have written the paper on this topic which is currently under the peer review process. 3) Together with Shallit we checked that a number x which has certain continued fraction expansion satisfies remarkable approximational properties. We wrote a paper on this topic which is accepted by Proc. of AMS journal. |

Exploitation Route | The immediate beneficiary of the results of my research is the academic society in general and more precisely such disciplines as analytical numbers theory, Diophantine approximation, Dynamical systems, measure and probability. Sets of badly approximable numbers and their analogues recently attract a lot of attention among these disciplines. |

Sectors | Other |

Description | As far as I know specialists in Diophantine approximation already use my results in their research. These results are available through the free arxiv repository as well as the Durham university online repository. Also they were shared in various conferences and during several research visits. Concerning the impact of my research out of the academy, the research I'm doing is purely theoretical. Historically this type of research provided the first step towards the production of the science intensive technology. So it has a high potential for long term impact. |

First Year Of Impact | 2014 |

Sector | Other |

Description | LMS Undergraduate Research Bursary for Matthew Northey |

Amount | £1,080 (GBP) |

Organisation | London Mathematical Society |

Sector | Learned Society |

Country | United Kingdom |

Start | 08/2015 |

End | 09/2015 |

Description | Collaboration with York |

Organisation | University of York |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | I make regular visits to the University of York where I contact people from the Number theory group, in particular Velani, Beresnevich, Zorin, Simmons and others. I discuss the mathematics with them which resulted in several research papers. From time to time I give talks at the number theory seminars there. Sometimes people from York also come to Durham to give talks. |

Collaborator Contribution | People from York share their knowledge with me and sometimes with my PDRA Harrap. This result in several publications already and hopefully will result in future publications. They also provide timeslots at their Number Theory seminars for the talks. Finally they provide a refund of my travel expenses after the First EPSRC grant came to its end. |

Impact | This collaboration resulted in several publications. The newest of them (since 2013) are Badziahin, Zorin: "Thue-Morse constant is not badly approximable" (IMRN) Badziahin, Velani: "Badly approximable points on planar curves and the problem of Davenport" (Math. Annalen) Badziahin, Zorin: "On generalised Thue-Morse functions and their values". (Preprint) Badziahin, Harrap, Nesharim, Simmons: "Topological games and Cantor-winning sets" (in preparation). |

Start Year | 2011 |

Description | Research visit of Dr German to Durham |

Organisation | Moscow State University |

Country | Russian Federation |

Sector | Academic/University |

PI Contribution | I invited Dr German to the university of Durham for a couple of weeks in April 2014. I wanted to use his expertise in the distribution of rational n-dimensional points to make some developments in the n-dimensional badly approximable points. Our collaboration was interesting, however it did not produce any research papers. |

Collaborator Contribution | Dr German shared his knowledge about n-dimensional rational points and their distribution. |

Impact | Mostly the transfer of the ideas between me and Dr German. |

Start Year | 2014 |

Description | Research visit to the University of Newcastle, Australia |

Organisation | University of Newcastle |

Department | School of Mathematics & Statistics |

Country | Australia |

Sector | Academic/University |

PI Contribution | I used the ideas from one of my papers about the Hausdorff dimension of well approximable points on planar curves in the inhomogeneous case to compute more sensitive Hausdorff s-measure of the same set. I also gave two talks at the university of Newcastle. One of them was at the "Number theory upside down" conference and another one was at the departmental seminar. |

Collaborator Contribution | Dr. Mumtaz Hussain shared his expertise to compute the Hausdorff s-measure of of well approximable points on planar curves in the inhomogeneous case. I also met with another mathematician from that university, Dr. M. Coons, his expertise helped me to develop my research of diophantine properties of certain automatic numbers like Thue-Morse number. |

Impact | It resulted in writing one paper "An inhomogeneous Jarnik type theorem for planar curves" which is submitted to Math. Proc. of Camb. Phil. Soc. journal. |

Start Year | 2014 |

Description | Research visit to the University of Waterloo, Canada |

Organisation | University of Waterloo |

Country | Canada |

Sector | Academic/University |

PI Contribution | I made a research visit to Prof. J. Shallit. I shared my ideas about the possible ways to attack p-adic Littlewood conjecture. I also used my expertise to help Jeffrey to solve the problem concerning the approximational properties of the number with certain continued fraction expansion. |

Collaborator Contribution | Prof. J. Shallit is a well known expert in combinatorics of words. During the research of the p-adic Littlewood conjecture I found that for potential counterexamples to the conjecture certain combinatorial conditions should be satisfied for their continued fraction expansion considered as an infinite word. Therefore Jeffries expertise was helpful for better understanding of the consequences of that conditions. |

Impact | It resulted in publishing one research paper "An unusual continued fraction" in Proc. of Amer. Math. Soc. journal. |

Start Year | 2015 |

Description | Easter School "Dynamics and Analytic Number Theory" |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Postgraduate students |

Results and Impact | I was a co-organiser of the Easter School "Dynamics and Analytic Number Theory" which was held at Durham University in 31 March - 4 April 2014. The aim of this event was to bring researchers from both areas of Mathematics together and to provide ample opportunities for communication. Around 60 young researches from around the world attended the school. 6 invited speakers gave a series of 2 - 3 hours minicourses. One more invited speaker gave an overview talk with the current state of the art and a list of open problems in the theory of Diophantine approximation. |

Year(s) Of Engagement Activity | 2014 |

URL | http://www.maths.dur.ac.uk/users/dzmitry.badziahin/2014_Easter_school/easter_index.html |