EPSRC Reference: 
EP/Y016769/1 
Title: 
The density of rational points near manifolds and applications 
Principal Investigator: 
Beresnevich, Professor V 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of York 
Scheme: 
Standard Research 
Starts: 
01 May 2024 
Ends: 
30 April 2027 
Value (£): 
442,630

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

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.
Rational points are everywhere dense, that is they are present in a neighborhood of any point, however small that neighborhood is. Nevertheless, the density of their distribution is not the same everywhere. Understanding the distribution of rational points often leads to major challenges. For instance, there are simple statements about the distribution of Farey fractions (rational numbers in the unit interval of bounded denominator) that are equivalent to Riemann Hypothesis  one of the biggest mathematical challenges.
This project aims to investigate the density of the distribution of rational points in small neighborhoods of manifolds, such as curves and surfaces, in ndimensional spaces. Since manifolds can be defined by systems of equations, the rational points in question arise as approximate solutions to these equations. Therefore the subject matter of this project is closely related to Diophantine equations and Diophantine geometry  two other mainstream areas of mathematics that deal with the existence and density of precise rational solutions to equations.
One of the central goals of this project is to investigate the density of rational points near generic manifolds by classifying and quantifying obstructions: small domains within manifolds that may result in abnormally high or abnormally low density. To achieve this goal we will develop a novel approach bringing together techniques and ideas from several areas including Diophantine approximation, the geometry of numbers and homogeneous dynamics.
The density of rational points near manifolds plays an important role in many other problems in number theory and other disciplines. 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. Already within this project we will consider applications of results on the density of rational points near manifolds to topical problems in number theory. Specifically we aim to resolve some outstanding problems on the spectrum of Diophantine exponents and multiplicative Diophantine approximations.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

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