Output Sum of Transducers: Limiting Distribution and Periodic Fluctuation

Abstract

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in [0, N ). Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework.Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.2010 Mathematics Subject Classification. 60F05; 68R15, 05A16, 68Q45, 11M41.

Asymptotic Analysis of Regular Sequences

“…We start with transducer automata. Transducers have been chosen in order to compare the results here with the previously available results [32]. In some sense, the results complement each other: While the results in [32] also contain information on the variance and the limiting distribution, our approach here yields more terms of the asymptotic expansion of the mean, at least in the general case.…”

“…These numbers are an example of an automatic sequence, therefore can be treated by a transducer. However, it turns out that the generic results (the results here and in [32]) degenerate: They are too weak to give a meaningful main term. Therefore a different effort is needed for esthetic numbers.…”

“…This is then used in the particular cases of esthetic numbers and in conjunction with the output sum of transducers. More specifically, in the latter example this yields the second order term in Corollary F; see also [32].…”

“…Transducers. The sum T (n) of the output labels of a complete deterministic finite transducer T when reading the q-ary expansion of an integer n has been investigated in [32]. As this can be seen as a q-regular sequence, we reconsider the problem in the light of our general results in this article; see Section 8.…”

“…The sequence of such numbers turns out to be q-automatic, thus are q-regular and can also be seen as an output sum of a transducer; see the first author's joint work with Kropf and Prodinger [32] or Section 8. However, the asymptotics obtained by using the main result of [32] is degenerated in the sense that the provided main term and second order term both equal zero; only an error term remains. On the other hand, using a more direct approach via our main theorem brings up the actual main term and the fluctuation in this main term.…”