Research

Preprints (Comments and mistake hunting are extremely welcome)

1. Counting rationals and diophantine approximation in missing-digit Cantor sets (With S. Chow and P. Varju)

2. Missing digits points near manifolds

3. On the absolute continuity of radial and linear projections of missing digits measures

4. Finer geometry of planar self-affine sets(With B. Bárány and A. Käenmäki)

5. Rational points near self-similar sets

In Press

1. Non-Salem sets in metric Diophantine approximation
To appear in IMRN(With K. Hambrook)
2. On the metric theory of multiplicative Diophantine approximation
To appear in Journal d'Analyse Mathématique
3. Fractal projections with an application in number theory
To appear in Ergodic Theory and Dynamical Systems

Publications

1. A note on dyadic approximation in Cantor's set
Indagationes Mathematicae Volume 34, Issue 1, January 2023, Pages 190-197(With D. Allen, S. Baker and S. Chow)
2. Dyadic Approximation in the Middle-Third Cantor Set
Selecta Mathematica volume 29, Article number: 11 (2023)(With D. Allen and S. Chow)
3. Times two, three, five orbits on T^2
Advances in Mathematics, Volume 399, 2022, 108289
4. Bernoulli convolutions with Garsia parameters in (1,sqrt{2}] have continuous density functions
Proc. Amer. Math. Soc. 150 (2022), 4359-4368
5. Dimensions of the popcorn graph
Proc. Amer. Math. Soc. 150 (2022), 4729-4742(With H-P Chen and J. Fraser)
6. Fourier decay of self-similar measures and self-similar sets of uniqueness
Anal. PDE 15 (3) 843 - 858, 2022.(With Peter Varju)
7. An improvement on Furstenberg's intersection problem
Transactions of the AMS, 374 (2021), 6583-6610
8. Additive properties of numbers with restricted digits
Algebra and Number Theory, 15, 2021, 1283-1301
9. Approximate arithmetic structure in large sets of integers
RAE, 46, 2021, 163-173(With J. Fraser)
10. Cube packings in Euclidean spaces
Mathematika 67 (2021), 288-295
11. On GILP's group theoretic approach to Falconer's distance problem
Glasg. Math. J. 63 (2021), 547-562
12. Lq-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums
Nonlinearity, Volume 34, Number 9(With J. Fraser, L. Lee and I. Morris)
13. On sets containing a unit distance in every direction
Discrete Analysis, 2021:5(With Pablo Shmerkin)
14. Digit expansions of numbers in different bases
Journal of Number theory, Volume 226, 2021, Pages 284-306(With S. Burrell)
15. On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result
Journal of Number Theory, Volume 224, 2021, Pages 243-273
16. Bernoulli decomposition and arithmetical independence between sequences
Ergodic theory and Dynamical systems, Volume 41(5), 2021, pp. 1590-1600
17. Weak tangents and level sets of Takagi functions
Monatshefte für Mathematik, Volume 192, 2020, pp. 249-264
18. On the Hausdorff dimension of microsets
Proceedings of the American Mathematical Society, Volume 147(11), 2019, pp. 4921-4936(With J.Fraser, D. Howroyd and A. Käenmäki)
19. Multi-rotations on the unit circle
Journal of Number theory, Volume 200, 2019, pp. 316-328
20. Dimension growth for iterated sumsets
Mathematische Zeitschrift, Volume 293, 2019, pp. 1015-1042.(With Jonathan Fraser and Douglas Howroyd)
21. Dimensions of triangle sets
Mathematika, Volume 65(2), 2019, pp. 311-332
22. A Fourier analytic approach to inhomogeneous Diophantine approximation
Acta Arithmetica, Volume 190, 2019, pp. 263-292
23. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra
Annales Academiæ Scientiarum Fennicæ Mathematica, Volume 44, 2019, pp. 379-387 (With J.Fraser, K.E.Hare, K.G.Hare,S.Troscheit)
24. Erdős Semi-groups, arithmetic progressions and Szemerédi's theorem
Real Analysis Exchange, Volume 44(1), 2019, pp. 101-118
25. Assouad dimension of random processes
Proceedings of the Edinburgh Mathematical Society, Volume 62(1), 2019, pp. 281-290 (With Douglas Howroyd)
26. New dimension spectra: finer information on scaling and homogeneity
Advances in Mathematics,Volume 329, Pages 273-328 (With Jonathan Fraser)
27. Dimensions of sets which uniformly avoid arithmetic progressions
International Mathematics Research Notices,rnx261 (With Jonathan Fraser and Kota Saito)
28. Arithmetic patches, weak tangents, and dimension
Bulletin of the London Mathematical Society,50 (2018) 85–95 (With Jonathan Fraser)
29. Root sets of polynomials and power series with finite choices of coefficients
Computational methods and Function theory (DOI 10.1007/s40315-017-0215-1) (With Simon Baker)
30. On generalized trigonometric functions and series of rational functions
Journal of Number Theory, Volume 180, 2017, pages 512-532
31. Assouad type spectra for some fractal families.
Indiana University Mathematics Journal, Volume 67(5), 2018, pp. 2005-2043 (With Jonathan Fraser)

Notes

Kakeya set is being defined in a rotational invariant way, so it should have some rotational symmetric properties. In fact, a certain rotational invariance of Kakeya set implies the Kakeya conjecture.

In this note, we prove one of Furstenberg's results on times 2, times 3 theorem for closed invariant sets