Dynamics of generalized Farey sequences with applications to equidistribution

The Farey sequence, discovered in the early nineteenth century, is now an important object in number theory, geometry, and homogeneous dynamics. For example, a conjecture concerning the distribution of Farey sequences is equivalent to the Riemann hypothesis, one of the most famous outstanding problems in number theory and, indeed, all of mathematics. It also plays a key role in approximation in number theory through its connections with objects called continued fractions and an important role in geometry and homogenous dynamics through its connections with the dynamics of horocycles, which are important curves in hyperbolic geometry. This leads to an elegant theory in number theory, geometry, and homogeneous dynamics whose link is the Farey sequence, whose applications are found in, for example, mathematical physics and applied dynamics, and whose influence reach out towards physics and biology.

The Farey sequence is easy to describe. Let n be a natural number. The Farey sequence of order n is a sequence of rational numbers in lowest terms between 0 and 1 with denominators less than or equal to n, ordered by increasing size. For example, the Farey sequence of order 1 is the sequence {0, 1} and the Farey sequence of order 2 is the sequence {0, 1/2, 1}. These sequences give rise, via hyperbolic geometry, to continued fractions which yield the best approximations of a given real number. Such approximations are a central concern in the subfield of number theory called Diophantine approximation. These sequences also are important in hyperbolic geometry itself because they help us understand horocycles and, in particular, how horocycles distribute under the dynamics of a natural flow, namely the geodesic flow. In this way, Farey sequences provide a deep link between number theory, homogeneous dynamics, and geometry, a link which should be generalised and deepened further to the benefit of all three fields.

This proposal aims to generalise and deepen this link, and the expected results will belong to three fields, multiplying their benefit. We will study the analog of the Farey sequence in very general settings such as spaces coming from locally compact Hausdorff (topological) groups and their discrete subgroups. These can be large and complicated spaces. Topological groups are spaces for which we have a notion of multiplication, namely any two elements multiplied together yields another element in the group, and which satisfy some sensible rules. Locally compact and Hausdorff are two topological notions and many interesting spaces, such as the plane, three-dimensional space or, more generally, manifolds, have these properties.

We will then use these generalized Farey sequences in two ways. The first is to study their number-theoretic properties in analogy with the classical Farey sequences and the rich number theory coming from them. The second is to use these sequences to study dynamics on these large and complicated spaces again in analogy with the classical Farey sequences and horocycles. Some of the tools that we will use are the mixing property coming from ergodic theory, Ratner's theorems coming from homogenous dynamics, Eisenstein series coming from analytic number theory, and harmonic analysis, which is a field of mathematics concerned with decomposing functions into the infinite sum of "wave-like functions.''

By generalising the elegant theory of which the Farey sequence is the link, we will also expand upon the applications and influences of the theory.

The proposed research is in pure mathematics and the primary impact will be academic, namely benefiting the fields of homogeneous dynamics, number theory, and geometry and the researchers working in those fields. Besides academic impact, there are other sources of indirect impact and these will be on training and on the effect that academic research has. This includes the effect the proposed research has on "delivering significant social and economic impact" according to the Bond report "The Era of Mathematics."

The indirect economic and societal impacts of research such as the proposed research has been documented in the Bond report. There are numerous industries, film, finance, medicine, engineering, and so on, that benefit from research in mathematics and these industries, in turn, provide benefits to society and employment for many people. Bond writes that Deloitte (2012) estimated the impact at "£200 billion annually to the UK economy in 2010, ... 2.8 million individuals in employment directly due to mathematical science research in the UK, a figure that is still rising, [and] 6.9 million individuals in employment due to the wider ripple effects of mathematical science research in the UK."

In particular, the three fields of the proposed research, homogeneous dynamics, number theory, and geometry, are all either part of or closely related to the field of mathematical analysis, which, in particular, has spectacular applications in physics and engineering. To name but one example, the field of signal processing is based largely upon the theory and applications of Fourier and harmonic analysis. Signal processing itself has many applications such as image processing, radar, sonar, and sensors. For example, efficient signal processing is integral to building self-driving cars because of the requirement for multiple sensor signals to be integrated and processed in real time, namely a real-time solution to the so-called "sensor fusion problem." Research in mathematical analysis, such as the proposed research, can, thus, have an indirect impact on society through contributing to the delivery of new products that people will use.

The impact of this proposal on training has many aspects. First, the proposal will benefit my own PhD student, whose studentship will span the grant period, by allowing him/her to interact with established and early career researchers at the conference that I plan to host at Exeter in August 2021. Second, the conference will benefit other students and early career researchers by allowing them to exchange ideas and interact with both established researchers and among themselves. Third, I will disseminate the expected results at the conference that I host and at other conferences, workshops, and seminars to help inspire the next generation of researchers in the interface of homogeneous dynamics, number theory, and geometry.

Thus new research in mathematics has important indirect impacts beyond academia.