Research Summary

Research Interest by Keywords

Metric Number Theory  ||  Diophantine Approximation  ||  Continued Fractions  ||  Dynamical Systems  ||  Ergodic Theory  ||  Iterated Function Systems  ||  Shrinking Target Properties  ||  Uniform Distribution of Sequences  || Discrepancy ||  Hausdorff Dimension  ||  Fractal Geometry  ||  Generalized Numeration System  ||  Non-Archimedean Spaces  ||  a-Adic Integers  ||  p-Adic Numbers  ||  Formal Laurent Series over a Finite Field  ||  Quasi-Monte Carlo Method || Exponential Sum 

Research Summary

My research focuses on the metrical theory of numbers. This is a branch of number theory that studies and characterizes sets of numbers from a measure-theoretic or probabilistic point of view. The central theme of this theory is to determine whether a given arithmetic property holds everywhere except on an exceptional set of measure zero. Also, metric number theory includes the study of the complexity of those exceptional sets in terms of Hausdorff dimension. Nowadays, the theory is deeply intertwined with measure theory, ergodic theory, dynamical systems, fractal geometry and other areas of mathematics. This has led my research to various related directions including subsequence ergodic theory, Diophantine approximation, uniform distribution theory of sequences, metric number theory in non-Archimedean settings and quasi-Monte Carlo method. Currently, I am venturing further afield to extend the lines of research, e.g. fractals and iterated function systems, metric Diophantine approximation and shrinking target properties, ergodic theory and analytic number theory, and applications of metric number theory and dynamical systems to applied science.

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Research Achievements

The following is a list of my research achievements. Note that the bracketed numbers are referred to my publications.

• • We proved in [1] the positive characteristic analogue of D.J. Rudolph's result regarding the famous H. Furstenberg's conjecture 2× and 3× invariant measure. Indeed, we showed that either the entropy of the maps is zero or the non-atomic measure is Haar measure. Moreover, in [13], we have recently improved our result by adjusting the condition that p and q are coprime to the condition that p does not divide any positive power of q.

  • In [4], we employed some geometric measure theory to settle the positive characteristic analogue of an open problem asked by R.D. Mauldin on the complexity of the set of Liouville numbers. Indeed, we could give a complete characterization of all Hausdorff measures of the set of Liouville numbers in the field of formal Laurent series over a finite field.

  • In [2] and [5], we used some subsequence ergodic theory to establish a generalized metric theory of continued fractions in both the settings of the p-adic numbers and the formal Laurent series over a finite field.

  • In [2] and [0], we developed new and much more powerful formulations of ergodicity and unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences.

  • We introduced in [3] the a-adic van der Corput sequence which significantly generalizes the classical van der Corput sequence. We also showed in [0] that it provides a wealth of low-discrepancy sequences which are very useful in the quasi-Monte Carlo method. Recently, we have successfully extended this one-dimensional sequence to the Halton sequence and the Hammersley point set in a generalized numeration system, called the Cantor expansion, in [9] and showed that it provides a wealth of low-discrepancy sequences by giving an efficient estimate of its discrepancy.

  • In [3] and [11], we used our subsequential characterization of unique ergodicity to solve the generalized version of an open problem asked by O. Strauch on the distribution of the sequence of consecutive van der Corput sequences.

  • Let P - 1 denote the set of primes minus 1. A classical theorem of A. Sárközy says that any set of natural numbers of positive density contains a pair of elements whose difference belongs to the set P - 1. In [6], we investigated the positive characteristic analogue of questions of this type, building on work of H. Furstenberg, by using an ergodic approach.

  • In [7], we used some ergodic methods to prove the uniform distribution of a large class of subsequences of a generalization of the classical Halton sequences. This built on earlier work of M. Hofer, M.R. Iacò and R. Tichy in the case of a generalization of the classical Halton sequences.

  • Quantitative versions of the central results of the metric theory of continued fractions were given primarily by C. de Vroedt. In [8], we were able to give improvements of the bounds involved by using a quantitative L2 ergodic theorem.

  • In [10], we sharpened some results in [5] by providing a quantitative version of the metric theory of continued fractions in the field of formal Laurent series over a finite field. We adapted Gál and Koksma's method for establishing the error term in the ergodic averages.

Research in Progress

Currently, I am working on the following problems.

• • Ψ-mixing property of continued fraction maps

  • Quantitative metric theory of continued fractions in positive characteristic

  • Exponential sums under periodic perturbation

  • Magnitude of the sum of partial quotients after multiplication