169 research outputs found

### Bounds for the discrete correlation of infinite sequences on k symbols and generalized Rudin-Shapiro sequences

Motivated by the known autocorrelation properties of the Rudin-Shapiro
sequence, we study the discrete correlation among infinite sequences over a
finite alphabet, where we just take into account whether two symbols are
identical. We show by combinatorial means that sequences cannot be "too"
different, and by an explicit construction generalizing the Rudin-Shapiro
sequence, we show that we can achieve the maximum possible difference.Comment: Improved Introduction and new Section 6 (Lovasz local lemma

### Remarks on separating words

The separating words problem asks for the size of the smallest DFA needed to
distinguish between two words of length <= n (by accepting one and rejecting
the other). In this paper we survey what is known and unknown about the
problem, consider some variations, and prove several new results

### Summation of Series Defined by Counting Blocks of Digits

We discuss the summation of certain series defined by counting blocks of
digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes
the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1}
s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2$. We recover this previous
result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J.
Number Theor

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