PhD Studentship: The Khintchine-Groshev Theorem: Beyond Removing Monotonicity
University of Exeter
This project is primarily in the field of Diophantine Approximation, which studies how well real numbers can be approximated by rational numbers and variations on this theme. Sets encountered in Diophantine Approximation are often fractal-like and it is natural to use notions from Fractal Geometry to study them. Diophantine Approximation and Fractal Geometry have found wide-ranging applications in areas such as medicine, materials science, geology, and wireless communications, and have attracted attention of sev...
This project is primarily in the field of Diophantine Approximation, which studies how well real numbers can be approximated by rational numbers and variations on this theme. Sets encountered in Diophantine Approximation are often fractal-like and it is natural to use notions from Fractal Geometry to study them. Diophantine Approximation and Fractal Geometry have found wide-ranging applications in areas such as medicine, materials science, geology, and wireless communications, and have attracted attention of sev...
Opportunity details
This project is primarily in the field of Diophantine Approximation, which studies how well real numbers can be approximated by rational numbers and variations on this theme. Sets encountered in Diophantine Approximation are often fractal-like and it is natural to use notions from Fractal Geometry to study them. Diophantine Approximation and Fractal Geometry have found wide-ranging applications in areas such as medicine, materials science, geology, and wireless communications, and have attracted attention of several Fields Medallists. A classical theorem of Khintchine (1924) states that the set of real numbers which can be approximated at a given rate by rationals has zero or full Lebesgue measure depending on whether a certain sum converges/diverges. This result relies on a monotonicity condition. Duffin and Schaeffer (1941) showed that this monotonicity assumption cannot be removed and conjectured the correct statement without monotonicity. Resolving the Duffin-Schaeffer Conjecture was one of the biggest unsolved problems in Diophantine Approximation until its resolution by Koukoulopoulos and 2022 Fields Medallist Maynard (2020). The Khintchine Groshev Theorem generalises Khintchine’s Theorem to higher dimensions and originally required monotonicity conditions. Beresnevich and Velani (2010) showed that monotonicity could be dropped from the homogeneous Khintchine-Groshev Theorem except when the result reduced to Khintchine’s Theorem. In a series of recent works ([Allen+Ramirez, 2023], [Hauke, 2025], [Kim, 2025], [Kim, 2026+], [Allen+Ramirez+Hauke-Treuer, 2026+]) it has been shown that in all except the classical one dimensional case (where monotonicity is known to be absolutely necessary), we can also remove monotonicity from the inhomogeneous Khintchine-Groshev Th...
Eligibility and requirements: Review the official provider page for the current applicant conditions and required documents.
Funding and benefits
Funding eligibility: UK Students, International Students. Funding amount: £21,805 The studentship will provide funding of fees and a stipend of £21,805 per annum for 2026/7..
How to apply
Diophantine Approximation and Fractal Geometry have found wide-ranging applications in areas such as medicine, materials science, geology, and wireless communications, and have attracted attention of several Fields Medallists. Possible directions for this project include: Applications of the “Independence Inheritance” phenomenon. However, this result is very general and one could investigate further applications of independence inheritance, both in Diophantine Approximation and beyond.
Open the official application information
Verify the deadline and final conditions on the official provider page before submitting an application.
Source and verification
Check the current official call before applying
Recorded source check: October 27, 2026. A stored check date does not guarantee that every field is complete or still current. Compare the provider, eligibility, funding and deadline with the live official call before applying.
Benefits
Funding eligibility: UK Students, International Students. Funding amount: £21,805 The studentship will provide funding of fees and a stipend of £21,805 per annum for 2026/7..
Eligibility summary
Requirements
Documents required
Language/test requirements
Listing-board disclaimer
Prime Scholarship Alerts is an independent listing board, not the scholarship provider, university or funder. We do not select applicants, collect provider application fees or guarantee an award. Verify requirements, deadlines, funding, fees and application instructions through the current provider page before applying.