# Details of Grant

EPSRC Reference: EP/X030784/1
Title: The Inhomogeneous Duffin-Schaeffer Conjecture
Principal Investigator: Beresnevich, Professor V
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of York
Scheme: Standard Research - NR1
Starts: 01 April 2023 Ends: 31 January 2024 Value (£): 80,984
EPSRC Research Topic Classifications:
 Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
 No relevance to Underpinning Sectors
Related Grants:
Panel History:
 Panel Date Panel Name Outcome 01 Dec 2022 EPSRC Mathematical Sciences Small Grants Panel December 2022 Announced
Summary on Grant Application Form
Diophantine approximation - the main concern of this project - is an area of number theory which, in simple terms, studies rational approximations to real numbers, that is approximations by fractions of two integers. It dates back to the ancient Greeks and Chinese who used good rational approximations to the number pi (=3.1415...) to predict the position of planets and stars. In 'everyday life' we often use truncated decimal expansions for the purpose, e.g. 3.14=314/100 to approximate pi. However, these are usually far from being good. For example, 22/7 uses fewer digits than 314/100 but is closer to pi, while 355/113 uses the same number of digits but accurately gives 5 decimal places of pi.

Good rational approximations are guaranteed by Dirichlet's fundamental theorem: for every irrational number x there are infinitely many rationals a/q approximating x to within 1 over the square of the denominator, q. Of course, individual real numbers, as opposed to all real numbers, may vary vastly in terms of how they can be approximated. For instance, Liouville numbers can be approximated infinitely often by rationals a/q to within 1 over any power of the denominator, while for badly approximable numbers that power can only be 2, as in Dirichlet's theorem. Metric number theory takes a probabilistic viewpoint and thus offers a middle ground between 'studying all' and 'studying individual' numbers. The central theme of this theory is to determine whether almost all or almost no real numbers can be approximated by rational numbers in a certain way.

In 1941 Duffin and Schaeffer stated a very general conjecture predicting how almost all (in probabilistic terms) real numbers can be approximated by rational numbers. Attempts to solve the conjecture have a long history and many discoveries along the way. The conjecture was eventually proved in a breakthrough by Koukoulopoulos and Maynard, which magnitude was recognised by a 2022 Fields Medal Award to Maynard. This project will investigate the far more general inhomogeneous version of the conjecture.

In inhomogeneous approximations the numerator of the rational number is shifted by a fixed real parameter - the inhomogeneous part. The reason for that is best described in terms of circle rotations. In the homogeneous case, if a/q approximates a real number x, any point on a given circle rotated q times by the angle alpha=2.pi.x returns to a neighborhood of its original position, which size is determined by the error of approximations. In the inhomogeneous case such rotations are used to hit the neighborhood of an arbitrary fixed point on the circle associated with the inhomogeneous part.

This project will develop a novel approach to the inhomogeneous Duffin-Schaeffer conjecture. In particular, we aim to discover the first irrational examples of the inhomogeneous part satisfying the conjecture. For its probabilistic nature the conjecture is unsurprisingly treated using a version of the second Borel-Cantelli lemma. This enables one to establish that a certain 'divergent' series of 'events' happens infinitely often with positive probability if we assume a ceratin independence of the events. Verifying the latter is the key to solving the problem and thus constitutes the core of this project. In particular, we will investigate how variations of the initial events can be used to improve existing and obtain new independence estimates. This will bring together techniques and ideas from Diophantine approximation, number theory and probability. In particular, we will develop novel tools in probability theory that will incorporate zero-one laws enabling one to extend positive to full probabilities. The theme of this project can be found in many other problems in number theory and other areas such as dynamical systems. Thus, we expect that the novel techniques and ideas that we will develop will have a lasting impact on the areas involved and far beyond.
Key Findings
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