We give an overview of recent results obtained by the applicant, as well as pointers to future research topics.
This includes theorems on sparse subsequences (such as subsequences indexed by arithmetic progressions with large common difference, the sequence of cubes, or the sequence of prime numbers) of certain morphic sequences.
These sequences are defined by a substitution on a finite alphabet, and appear in areas as diverse as computer science, or the theory of dynamical systems.
In particular, we are led to the important notion of the level of distribution of a sequence, which is a fruitful concept also in the context of digital expansions.

The classical base-q expansion of an integer is closely connected to the question of divisibility of binomial coefficients.
In particular, the 2-adic valuation (the exponent of 2 in the prime factor decomposition) of binomial coefficients is not completely understood yet.
We will consider this case more closely, and point out recent results, and open problems.