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Details of Grant 

EPSRC Reference: EP/M021858/1
Title: Diophantine properties of Mahler's numbers
Principal Investigator: Zorin, Dr E
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of York
Scheme: First Grant - Revised 2009
Starts: 01 July 2015 Ends: 30 June 2017 Value (£): 93,505
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
Related Grants:
Panel History:
Panel DatePanel NameOutcome
03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
Diophantine approximation is a subfield of Number Theory which is mainly concerned with the quantitative aspects of approximation of real numbers by rational fractions. Today, the theory is deeply intertwined with other areas of mathematics such as ergodic theory, dynamical systems, fractal geometry, commutative algebra and algebraic geometry. It also has links to theoretical computer science and theory of differential equations. At the same time, Diophantine approximation continues to play a significant role in applications to real world problems including those arising from the rapidly developing areas of electronic communications, antenna design and signal processing.

The main aim of this research project is to investigate Diophantine properties of generalized Mahler numbers. The project contains genuine aspects of interdisciplinarity via (i) the interaction between Diophantine approximation properties of automatic numbers and wireless communication, and (ii) the interaction between Mahler numbers and the long-standing Hartmanis-Stearns conjecture in computer science.

One of the main goals of the proposed research is to develop a general criterion that enables one to determine whether or not a given Mahler number is badly approximable or not. This goal is naturally connected to the task of determining the irrationality exponent of a given Mahler number. At present, the techniques available are rather limited and existing results are established on a case by case basis. A general criterion would be highly desirable. Recent publications discovered interesting possible approach to this problem, which still needs to be developed.

Continuing with the theme of irrationality exponent, another key goal of the research is to develop a general criterion that guarantees a given generalized Mahler number has a finite irrationality exponent. This avenue of research will be further extended to determine the class of generalized Mahler numbers that do not belong to the class U of Mahler's classification of real numbers.

Another aim of the proposed programme is a well-known conjecture that Mahler numbers are either rational or transcendental. This conjecture is important from the point of view of the general transcendence theory, and also because it has links to theoretical computer science. The proposed research will seek a complete resolution of this conjecture and indeed investigating its analogue for generalised Mahler numbers.

An important part of this project is to build upon the existing collaboration with engineers to find and solve theoretical questions emerging from signal processing. This is aimed to facilitate and accelerate the development of models favouring the emergence of new technologies in wireless communication.

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Organisation Website: http://www.york.ac.uk