397 research outputs found
Complexity of Leading Digit Sequences
Let denote the sequence of leading digits of in base . It
is well known that if is not a rational power of , then the sequence
satisfies Benford's Law; that is, digit occurs in with
frequency , for .
In this paper, we investigate the \emph{complexity} of such sequences. We
focus mainly on the \emph{block complexity}, , defined as the
number of distinct blocks of length appearing in . In our main
result we determine for all squarefree bases and all
rational numbers that are not integral powers of . In particular, we
show that, for all such pairs , the complexity function is
\emph{affine}, i.e., satisfies for all
, with coefficients and , given explicitly in
terms of and . We also show that the requirement that be squarefree
cannot be dropped: If is not squarefree, then there exist integers with
for which is not of the above form.
We use this result to obtain sharp upper and lower bounds for ,
and to determine the asymptotic behavior of this function as
through squarefree values. We also consider the question which linear functions
arise as the complexity function of some leading digit
sequence .
We conclude with a discussion of other complexity measures for the sequences
and some open problems
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