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Schmidt games and Cantor winning sets (2024)
Journal Article
Badziahin, D., Harrap, S., Nesharim, E., & Simmons, D. (online). Schmidt games and Cantor winning sets. Ergodic Theory and Dynamical Systems, https://doi.org/10.1017/etds.2024.23

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We su... Read More about Schmidt games and Cantor winning sets.

A problem in non-linear Diophantine approximation (2018)
Journal Article
Harrap, S., Hussain, M., & Kristensen, S. (2018). A problem in non-linear Diophantine approximation. Nonlinearity, 31(5), 1734-1756. https://doi.org/10.1088/1361-6544/aaa498

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations... Read More about A problem in non-linear Diophantine approximation.

Cantor-winning sets and their applications (2017)
Journal Article
Badziahin, D., & Harrap, S. (2017). Cantor-winning sets and their applications. Advances in Mathematics, 318, 627-677. https://doi.org/10.1016/j.aim.2017.07.027

We introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt’s (α, β)-game: these include maximal Hausdorff dimension, invariance under countable intersections wit... Read More about Cantor-winning sets and their applications.

An Inhomogeneous Jarník type theorem for planar curves (2016)
Journal Article
Badziahin, D., Harrap, S., & Hussain, M. (2017). An Inhomogeneous Jarník type theorem for planar curves. Mathematical Proceedings of the Cambridge Philosophical Society, 163(1), 47-70. https://doi.org/10.1017/s0305004116000712

In metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental t... Read More about An Inhomogeneous Jarník type theorem for planar curves.

A note on weighted badly approximable linear forms (2016)
Journal Article
Harrap, S., & Moshchevitin, N. (2017). A note on weighted badly approximable linear forms. Glasgow Mathematical Journal, 59(2), 349-357. https://doi.org/10.1017/s0017089516000203

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from... Read More about A note on weighted badly approximable linear forms.

A note on badly approximabe sets in projective space (2016)
Journal Article
Harrap, S., & Hussain, M. (2017). A note on badly approximabe sets in projective space. Mathematische Zeitschrift, 285(1), 239-250. https://doi.org/10.1007/s00209-016-1705-y

Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result... Read More about A note on badly approximabe sets in projective space.