Research

My main areas of research are dynamical systems, Diophantine approximation, and the interactions between the two.

Below is a list of all my papers which are either published or available on arXiv. Alternatively, here is a list of my papers on arXiv.

Publications and preprints:

61. Hausdorff dimension of differences of badly approximable sets (joint with D. Hatefi), preprint 2024. arXiv

60. Shrinking targets versus recurrence: the quantitative theory (joint with J. Levesley, B. Li, and S. Velani), preprint 2024. arXiv

59.  Exact dimensions of the prime continued fraction Cantor set  (joint with T. Das), preprint 2023. arXiv

58.  Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions (joint with A. Bakhtawar), Res. Number Theory 9, no.54 (2023)  arXiv | journal version

57. The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers (joint with P. Bos and M. Hussain), Proc. Amer. Math. Soc. 151, no. 5 (2023), 1823–1838. arXiv | journal version

56. Diophantine approximation on curves (joint with M. Hussain and J. Schleischitz), preprint 2019, to appear in Algebra Number Theory. arXiv

55. Dynamical Borel-Cantelli lemma for recurrence theory (joint with M. Hussain, B. Li, and B. Wang), Ergodic Theory Dynam. Systems 42, no. 6 (2022),1994-2008 . arXiv | journal version

54. Metrical systems of affine forms (joint with M. Hussain and S. Kristensen), J. Number Theory 213 (2020), 67–100. arXiv  | journal version

53. Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism (joint with T. Das, L. Fishman, and M. Urbański), Sel. Math. New Ser. 29, no. 19 (2023). arXiv | journal version

52. The measure game (joint with L. Crone, L. Fishman, S. Jackson, and H. Schuerger), preprint 2019. arXiv

51. On the dimension spectra of infinite iterated function systems (joint with T. Das), preprint 2019. arXiv

50. Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images (joint with Á. Farkas, J. Fraser, and E. Nesharim), Mathematika 67, no. 1 (2021), 196-213. arXiv | journal version

49. A variational principle in the parametric geometry of numbers (joint with T. Das, L. Fishman, and M. Urbański), Adv. Math. 437 (2024) 109435. arXiv | journal version

48. Naming the largest number: exploring the boundary between mathematics and the philosophy of mathematics, preprint 2018. arXiv

47. Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents (joint with S. Chow, A. Ghosh, L. Guan, and A. Marnat), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXI (2020), 643-671. arXiv | journal version

46. A general principle for Hausdorff measure (joint with M. Hussain), Proc. Amer. Math. Soc. 147, no.9 (2019), 3897–3904. arXiv | journal version

45. Schmidt games and Cantor winning sets (joint with D. Badziahin, S. Harrap, and E. Nesharim), Ergodic Theory Dynam. Systems 45, no.1 (2025) 71-110. arXiv | journal version

44. The generalised Baker–Schmidt problem on hypersurfaces (joint with M. Hussain and J. Schleischitz), Int. Math. Res. Not. IMRN 2021, no.12 (2019) 8845–8867. arXiv | journal version

43. Intersecting limit sets of Kleinian subgroups and Susskind's conjecture (joint with T. Das), Proc. Amer. Math. Soc. 148, no. 8 (2020) 3203–3207.  arXiv | journal version

42. A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation (joint with T. Das, L. Fishman, and M. Urbański), C. R. Math. Acad. Sci. Paris 355 (2017), no. 8, 835–846. arXiv | journal version

41. Quantitative results using variants of Schmidt's game: Dimension bounds, arithmetic progressions, and more (joint with R. Broderick and L. Fishman), Acta Arith. 188 (2019), no. 3, 289–316 . arXiv | journal version

40. The gap of uncertainty (joint with H. Yu and A. Zafeiropoulos), appendix to Marstrand's theorem revisited: projecting sets of dimension zero (V. Beresnevich, K. Falconer, S. Velani, and A. Zafeiropoulos), J. Math. Anal. Appl. 472 (2019), no. 2, 1820–1845. arXiv | journal version

39. The Hausdorff measure version of Gallagher's theorem -- closing the gap and beyond (joint with M. Hussain), J. Number Theory 186 (2018) 211-225. arXiv | journal version

38. Random walks on homogeneous spaces and Diophantine approximation on fractals (joint with B. Weiss), Invent. Math. 216 (2019), no. 2, 337–394. arXiv | journal version

37. Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points (joint with V. V. Beresnevich, A. Ghosh, and S. L. Velani), J. Lond. Math. Soc. (2) 98 (2018), no. 2, 306–328.  arXiv | journal version

36. Badly approximable points on self-affine sponges and the lower Assouad dimension (joint with T. Das, L. Fishman, and M. Urbański), 39 (2019), no. 3, 638–657.  arXiv | journal version

35. Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals (joint with L. Fishman and K. Merrill), Ann. Comb. 22 (2018), no. 2, 271–293 . arXiv | journal version

34. The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result (joint with T. Das), Invent. Math. 210 (2017), no. 1, 85–134. arXiv | journal version

33. Badly approximable vectors and fractals defined by conformal dynamical systems (joint with T. Das, L. Fishman, and M. Urbański), Math. Res. Lett. 25 (2018), no. 2, 437–467  arXiv | journal version

32. Some manifolds of Khinchin type for convergence, J. Théor. Nombres Bordeaux 30 (2018), no. 1, 175–193. arXiv | journal version

31. The Hurwitz continued fraction expansion as applied to real numbers (expository paper), Enseign. Math. 62 (2016), no. 3-4, 475–485. arXiv | journal version | review

30. A proof of the matrix version of Baker's conjecture in Diophantine approximation (joint with T. Das), Math. Proc. Cambridge Philos. Soc. 167 (2019), No. 1, 159–169. arXiv | journal version

29. Rational approximation of affine coordinate subspaces of Euclidean space (joint with F. A. Ramirez and F. Suess), Acta Arith. 177 (2017), no. 1, 91–100.  arXiv | journal version

28. A Hausdorff measure version of the Jarnik–Schmidt theorem in Diophantine approximation, Math. Proc. Cambridge Philos. Soc. 164 (2018), no. 3, 413–459. arXiv | journal version

27. Extremality and dynamically defined measures, II: Measures from conformal dynamical systems (joint with T. Das, L. Fishman, and M. Urbański), Ergodic Theory Dynam. Systems 41, no.8 (2021) 2311-2348. arXiv | journal version

26. Decaying and non-decaying badly approximable numbers (joint with R. Broderick and L. Fishman), Acta Arith. 177 (2017), no. 2, 143–152. arXiv | journal version | review

25. New examples of complete sets, with connections to a Diophantine theorem of Furstenberg (joint with V. Bergelson), Acta Arith. 177 (2017), no. 2, 101–131. arXiv | journal version | review

24. Extremality and dynamically defined measures, I: Diophantine properties of quasi-decaying measures (joint with T. Das, L. Fishman, and M. Urbański), Selecta Math. 24 (2018), no. 3, 2165–2206  arXiv | journal version

23. Dimension rigidity in conformal structures (joint with T. Das and M. Urbański), Adv. Math. 308 (2017), 1127–1186. arXiv | journal version | review

22. The Banach–Mazur–Schmidt game and the Banach–Mazur–McMullen game (joint with L. Fishman and V. Reams), J. Number Theory 167 (2016), 169–179. arXiv | journal version | review

21. Variations on Dirichlet's theorem (joint with L. Fishman), J. Number Theory 162 (2016), 11–22. arXiv | journal version | review

20. Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces (joint with L. Fishman and K. Merrill), Selecta Math. (N.S.) 24 (2018), no. 5, 3875–3888. arXiv | journal version

19. An analogue of a theorem of Kurzweil, Nonlinearity 28 (2015), no. 5, 1401–1408. arXiv | journal version | review

18. Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions (joint with L. Fishman, B. Mance, and M. Urbański), Bull. Aust. Math. Soc. 92 (2015), no. 2, 205–213. arXiv | journal version | review

17. Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind (joint with L. Fishman and M. Urbański), Geom. Dedicata 178 (2015), 95–101. arXiv | journal version | review

16. Geometry and dynamics in Gromov hyperbolic metric spaces I: with an emphasis on non-proper settings (joint with T. Das and M. Urbański), Mathematical Surveys and Monographs, 218. American Mathematical Society, Providence, RI, 2017. arXiv | AMS version

16b. Tukia's isomorphism theorem in CAT(-1) spaces (split from #16), Ann. Acad. Sci. Fenn. Math. 41 (2016), 659–680. arXiv | journal version | review

15. A Danzer set for axis parallel boxes (joint with Y. Solomon), Proc. Amer. Math. Soc. 144 (2016), no. 6, 2725–2729. arXiv | journal version | review

14. On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2675–2686. arXiv | journal version | review

13. Extrinsic Diophantine approximation on manifolds and fractals (joint with L. Fishman), J. Math. Pures Appl. (9) 104 (2015), no. 1, 83–101. arXiv | journal version | review

12. Intrinsic Diophantine approximation on manifolds (joint with L. Fishman, D. Kleinbock, and K. Merrill), preprint 2014.

12a.  Intrinsic Diophantine approximation on manifolds: General theory (split from #12), Trans. Amer. Math. Soc. 370 (2018), no. 1, 577–599. arXiv | journal version | review

12b. Intrinsic Diophantine approximation on quadric hypersurfaces (split from #12), J. Eur. Math. Soc. 24 (2022), no. 3, pp. 1045–1101 . arXiv | journal version

11. Relative equilibrium states and dimensions of fiberwise invariant measures for random distance expanding maps (joint with M. Urbański), Stoch. Dyn. 14 (2014), no. 1. journal version | review

10. Unconventional height functions in simultaneous Diophantine approximation (joint with L. Fishman), Monatsh. Math. 182 (2017), no. 3, 577–618. arXiv | journal version | review

9. Bad(s,t) is hyperplane absolute winning (joint with E. Nesharim), Acta Arith. 164 (2014), no. 2, 145–152. arXiv | journal version | review

8. Determinacy and indeterminacy of games played on complete metric spaces (joint with L. Fishman and T. Ly), Bull. Aust. Math. Soc. 90 (2014), no. 2, 339–351. arXiv | journal version | review

7. Diophantine approximation in Banach spaces (joint with L. Fishman and M. Urbański), J. Théor. Nombres Bordeaux 26 (2014), no. 2, 363–384. arXiv | journal version | review

6. Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (joint with L. Fishman and M. Urbański), Mem. Amer. Math. Soc. 254 (2018), no. 1215, arXiv | journal version

5. Badly approximable systems of affine forms and incompressibility on fractals (joint with R. Broderick and L. Fishman), J. Number Theory 133 (2013), no. 7, 2186–2205. arXiv | journal version | review

4. Intrinsic approximation for fractals defined by rational iterated function systems – Mahler’s research suggestion (joint with L. Fishman), Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 189–212. arXiv | journal version | review

3. Diophantine properties of measures invariant with respect to the Gauss map (joint with L. Fishman and M. Urbański), J. Anal. Math. 122 (2014), 289–315. arXiv | journal version | review

2. Random iteration of rational functions, Ph.D. thesis, University of North Texas, 2012. arXiv

1. Conditional measures and conditional expectation; Rohlin’s disintegration theorem, Discrete Contin. Dyn. Syst. 32 (2012), no. 7, 2565–2582. journal version | review