Correlations of k-regular and morphic sequences

The present project is concerned with digit representations of natural numbers. The base 10- expansion used in everyday life, and the nowadays equally important binary expansion are examples of such representations. A fundamental question, which, surprisingly, is not completely understood, is the following: in what manner do the digits of a number change when a constant is added? For example, it is conjectured that for most numbers, the number of ones in its binary expansion does not decrease when a constant is added. This seemingly trivial example is representative for a class of unsolved problems that are very easy to formulate, but apparently very difficult to prove. Moreover, these problems exhibit connections to various areas of mathematics. For example, the above-mentioned question, due to T. W. Cusick, is relevant in cryptography, since its answer would have effects on the construction of codes with desirable cryptographic properties. On closer inspection we see close connections to questions of a different type: what does the binary expansion (for example) of prime numbers look like? Only in the recent past it was proved by Mauduit and Rivat that one half of the prime numbers has an even number of ones in binary, while the other half has an odd number. In this formulation, half has a precise mathematical meaning (which of course has to be defined). Generalizing this theorem of certain substitutive sequences is an important goal of this project. As an example of such a result we note the preprint Primes as Sums of Fibonacci Numbers (jointly with M. Drmota and C. Müllner), where prime numbers are represented, not as a sums of powers of a natural number (as in the Mauduit--Rivat result), but as sums of Fibonacci numbers.This article forms part of the study of general substitutive sequences along the sequence of prime numbers, which is a vast, unexplored area, exhibiting important connections to the theory of dynamical systems. Summarizing, we wish to push forward solutions of number-theoretic problems that are non- artificial, elegant, and very easy to formulate, but difficult to prove, and combining methods coming from diverse areas of mathematics.