Wolfgang Steiner: Central Limit Theorems for Digital Expansions

The statistical behaviour of digital expansions plays a role in the analysis of several algorithms (divide-and-conquer strategies such as mergesort, traffic cutback, determination of sets with maximal number of interconnecting edges, RSA, ...). For example, it is well known that the sum-of-digits function $s_q(n)$ with respect to an integer base $q$ satisfies a central limit theorem on the set of integers $n<N$ with mean and variance proportional to $\log N$. (Interestingly this is also true on the set of prime numbers $p<N$ as well as on polynomial sequences.)

The purpose of this talk is to demonstrate a general method for determining limiting distributions of that kind which also works for more general digital expansions, e.g. those with respect to a linear recurrence such as the Zeckendorf expansion (based on the Fibonacci numbers). The problem consists mostly in determining the value of a digit without using the greedy algorithm. For a large class of recurrences, this can be done by using Rauzy fractals. Finally, the central limit theorem is then an (almost automatic) consequence of estimates for exponential sums.

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